Math Master Set

Implication

"P implies Q" written P=>Q false when P is true and Q is false. True otherwise.

Difference

A-B = {x: xeA and x~eB}

codomain of f: A→B

B

|x| proof

Divide into cases

Relatively Prime

GCD(a, b) = 1

What makes two sets equal? Give an example

If they contain the exact same elements. {2,4,6,8} = {8,6,4,2}(pg. 3)

What is an identity function?

It's outputs are it's inputs. e.g. y=x

Identity Function

Let A be a set. Define the function 1A : A → A by 1A(a) = a for all a ∈ A. This is the identity function on A.

Composite

Let a be an integer > 1. There are positive integers b and c with 1 < b,c < a such that a=bc.

What is an injective function?

Let f: A -> B be a function. We say f is injective if each element of B is the second coordinate of at most one ordered pair in f.

Binomial Coefficient

Let n, k in the integers. (n choose k) = {n!/(k!(n-k)!) if k is in [0, n], 0 if otherwise.

N

Natural Numbers {1,2,3,...}

Proving set equality you must...

Prove that A is a subset of B and B is a subset of A, therefore they are equal

Come up with a logically equivalent statement for P⇒Q

P⇒Q=(~Q)⇒(~P) pg. 49

Q

Rational Numbers

Indexed sets

Sets that are defined by an index, I. e.g. A_1, A_2, A_3,...,A_n is the sets with index set I={1,2,3,...,n} (pg. 24)

Divisibility

So, a|b if for some k in the integers, (ak=b) or (b/a)

partition of a set Ex. A= {1,2,3,4,5,6}

S₁= {{1,2},{5},{3,6,4}}

If we know that some uncountable set U is a subset of another set A, what do we know about A?

That A is uncountable (Theorem 13.9 pg. 331)

Contrapositive form of P⇒Q vs. the converse of P⇒Q.

The contrapositive form of P⇒Q is ~Q⇒~P and is logically equivalent to P⇒Q. The converse of P⇒Q is Q⇒P which is not logically equivalent to P⇒Q (pg. 102)

Ways to express P⇒Q

These all can be replaced with, "If P, then Q" pg. 43

What does "Let f: A -> B be a function" mean?

To specify that f is a function from A to B

Rational numbers

Two integers that divide

Transversal

When a set S as a subset of A exists, such that S contains exactly one representative of every equivalence class of the relation.

What does it mean for statements to be equivalent? How do we prove equivalent statements?

When a theorem asserts that a list of statements is equivalent, it is asserting that the statements are all true or they are all false. We prove equivalent statements using a sort of logical chain.

domain of f: A→B

dom f = A

Union and intersection of an INFINITE number of sets

pg. 25

Do questions 1.8:9,13

pg. 28

Natural numbers

ℕ={1,2,3,4,5...} (positive whole numbers pg. 6) Sometimes zero is included in the natural numbers

Why is the empty set a subset of every set? That is, ∅⊆B for any set B?

∅⊈B for any set B means there must be at least one element in ∅ that isn't in B. This clearly is not true because ∅ has no elements therefore ∅⊆B for any set B (fact 1.2 pg. 11)

and

∧

Disjunction

"P or Q" written PVQ. True if one or both P and Q are true

Associative Laws

(1) A U (B U C) = (A U B) U C (2) A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive Laws

(1) A U (B ∩ C) = (A U B) ∩ (C U A) (2) A ∩ (B U C) = (A ∩ B) U (C ∩ A)

De Morgan's Laws

(1) A U B (line over whole thing) = A ∩ B (lines over individual letters) (2) A ∩ B (line over whole thing) = A U B (lines over individual letters)

Commutative Laws

(1) A U B = B U A (2) A ∩ B = B ∩ A

Start a contradiction proof (P implies Q)

(1) assume P is true and Q is false find a contradiction (2) assume (P implies Q) is false find a contradiction

If we have n different things. How many length-k lists can we make if (A) repetition is allowed? (B) repetition is not allowed?

(A) n^k lists (B) n(n-1)(n-2)...(n-k+1)

The sum of any two integers is an integer. The difference of any two integers is an integer. and the product of any two integers in an integer

(a,b∈ℤ)⇒((a+b∈ℤ)∧(a-b∈ℤ)∧(ab∈ℤ)) (pg. 91)

Example of a constructive proof

(pg. 128)

The Binomial Theorem

(x+y)^n = ∑₀ (n choose k)x^(n-k)y^k

The Binomial Theorem

(x+y)^n= sigma( (n choose k)(x^(n-k))(y^k))

What is an ordered triple?

(x,y,z) (pg. 9)

injective function : A → B

***ONE-TO-ONE (V x₁,x₂∈A)(x₁≠x₂ => f(x₁)≠f(x₂)) or (V x₁,x₂∈A)(f(x₁)=f(x₂) => x₁=x₂)

surjective function ƒ: A → B

***ONTO (V z∈B)(∃ x∈A)(g(x)=z)

Fundamental Theorem of Arithmetic

*There exists a unique factorization of n.* Let n > 1. and n is an integer. Then n has a factorization into primes n = (p1)(p2)...(pk) with p1<=p2<=...<=pr, and this factorization is unique.

Fundamental Theorem of Arithmetic

*There exists a unique factorization of n.* Let n > 1. and n is an integer. Then n has a factorization into primes n = (p1)(p2)...(pk) with p1≤p2≤...≤pr, and this factorization is unique.

What is nC0 where n is a natural number?

1

What is nCn where n is a natural number?

1

What are the three categories of statements?

1) All the statements that have been proven to be true (theorems, propositions, lemmas, etc.) 2) Conjectures (statements whose truth has not been determined) 3) false statements (pg. 147)

1) Sets A and B have the same cardinality if... 2) The naturals and integers have the same/different cardinality because... 3) The naturals and reals do/don't have the same cardinality because...

1) if and only if there exists a bijection A→B 2) The same cardinalities because there exists a bijection between them. 3) Do not have the same cardinality because there exists no bijection between them.

What is the e Fibonacci sequence?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Let F(n) be the nth term of the sequence where n≥3. F(n)=F(n-2)+F(n-1). In other words, we let the first two terms be 1 and get the nth term by summing the previous two terms.

Find GCD and LCM with matrix and the linear combination

1. 0. 391 0. 1. 492 Continue to do -1 or -3 or whatever Whatever is left and not 0 is GCD The numbers times the zero part is LCM

six surprises about cardinality

1. There are different sizes of infinity. 2. |Z| = |N| 3. |Q| = |N| 4. |Q| < |R| 5. There is no largest infinity. 6. It is an undecidable question whether there is a cardinality between |N| and |R|

True/false 1. (∀x ∈ A)(∃y ∈ A)(x+y is even) 2. (∃y ∈ A)(∀x ∈ A)(x+y is even) A= {1,2,3}

1. true 2. false

What's the smallest prime number?

2

List the first few prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89

What is the length of the list (0,(0,1,1))?

2. This list has entries of 0 and another list: (0,1,1) (pg. 64)

How many letters are in the alphabet? How many digits are there?

26 letters. 10 digits.

Cardinality of the power set

2^n where n is the cardinality of the set

A ∩ C (line over C)

= A - C

Relation

A and B are two sets. A relation from A to B is a subset of AxB. That is, R is a set of ordered pairs where the first coordinate belongs to A and the second belongs to B.

Bijective fucntion

A bijective function is a function that is both injective and surjective.

What is a bijective function?

A bijective function is a function that is both injective and subjective.

What is a bijection?

A bijective function. I.e. a function that is both injective and surjective

Proper Subset

A c B if A_c_B and A~= B

Partition

A collection of subsets of A where the collections contain every element of A collectively, but no elements are in two subsets, and no empty set

What is a set?

A collection of things. These things are called the elements of the set and they can be just about anything (pg. 3)

Define a composite number

A composite number is an integer c>1 such that c is not prime. I.e. there is at least one positive divisor of c that is greater than 1 but is less than c (pg. 262)

Tautology

A compound statement S is a tautology if it is true for all possible values and combinations of statements that make up S.

Statement

A declarative sentence that is true or false (but not both).

What is the multiplication principle?

A fast way of counting (pg. 65)

Theorem 18.12

A fast way to compute the GCD.

surjective function

A function f : A → B is surjective if every element of B appears at least once as the second coordinate of an ordered pair in f.

Define an injective function

A function f: A→B is injective if for every x,y∈A, x≠y implies f(x)≠f(y). It means the function is one to one. In essence, injective means that unequal elements in A always get sent to unequal elements in B

Define a surjective function

A function f: A→B is surjective if for every b∈B there is an a∈A with f(a)=b (pg. 201) Surjective means that every element of B has an arrow pointing to it, that is, it equals f (a) for some a in the domain of f Note: a function is surjective if and only if its codomain equals its range (pg. 202)

Bijective

A function f:A-->B is called bijective if it is both one-to-one and onto.

Surjective (onto)

A function f:A-->B is onto if every element of the codomain B is the image of some element of A. (aka every element in B has something mapped to it)

Function

A function from A to B, written f: A-->B is a relation from A to B where every element of aeA is the first coordinate of exactly one ordered pair in f. (vertical line test basically)

Define a bijective function

A function is bijective if it is both injective and surjective (pg. 201).

Proper Set

A is a subset of S, but not equal to S

Congruents

A is congruent to b mod n is equal to n|a-b

What does A⊈B mean?

A is not a subset of B. It means there is at least one element in A that's not in B. We could also say that it is NOT true that every element of A is an element of B (pg. 11)

What is domain in "Let f: A -> B be a function"

A is the domain of f

What is a non-repetitive list?

A list in which repetition is not allowed (pg. 66)

Define theorem

A mathematical statement that is true and can be (and has been) verified as true (pg. 87)

Definition of prime

A natural number n is prime if it has exactly two positive divisors, 1 and n. So 2 is prime and 1 is not

Well-Ordered

A non-empty set S of real numbers is well-ordered if every non-empty subset of S has a least element.

well-ordered

A nonempty set S of real numbers is said to be well-ordered if every nonempty subset of S has a least element.

Definition of a perfect number

A number p∈ℕ is perfect if it equals the sum of its positive divisors less than itself. (e.g. 6=1+2+3, 28=1+2+4+7+14)

Irrational

A number that cannot be written as a fraction (Root 2, or Root 3)

Partition

A partition of A is defined as a set P of subsets of A such that: 1. None of the sets in P are empty i.e empty set is not an element of P 2. The union of sets in P is A 3. The intersection of two distinct sets in P is the empty set

Define partition.

A partition of a set A is a set of non-empty subsets of A, such that the union of all the subsets equals A, and the intersection of any two different subsets is ∅. Intuitively, a partition is just a dividing up of A into pieces (pg. 189)

Define a prime number

A prime number is an integer p>1 such that the only positive divisors of p are 1 and p (Definitino 34.1 pg. 262)

Mathematical Induction

A proof technic where you know that p(1) is true and assume if p(k) is true then p(k+1) is true.

Mathematical Induction

A proof technique where you know that p(1) is true (base case) and assume that p(k) is true and show that p(k+1) is true (inductive step)

Rational Number

A real number that can be expressed in the form m/n, where m, n∈Z and n = 0, is called a rational number.

Define rational and irrational

A real number x is rational if x=a/b for some a,b∈ℤ. Also, x is irrational if it is not rational that is if x≠a/b for EVERY a,b∈ℤ

reflexive

A relation R defined on a set A is called reflexive if xRx for every x∈A

Symmetric

A relation R defined on a set A is called symmetric if whenever xRy, then yRx for all x, y∈A.

symmetric

A relation R defined on a set A is called symmetric if whenever xRy, then yRx for all x,y ∈ A

Transitive

A relation R defined on a set A is called transitive if whenever xRy and yRz, then xRz for all x, y,z∈A.

transitive

A relation R defined on a set A is called transitive if whenever xRy and yRz, then xRz for all x,y,z ∈ A

What is an equivalence relation? Give some examples

A relation R on a set A is an equivalence relation if it is reflexive, symmetric and transitive. The "is equal to", "has the same parity as", "has the same sign as", and "has the same parity and sign as" relations are all equivalence relations. (pg. 184)

Equivalence Relation

A relation R on a set A is called an equivalence relation if R is reflexive, symmetric and transitive.

equivalence relation

A relation R on a set A is called an equivalence relation if R is reflexive, symmetric, and transitive.

Formal definition of relation from a set A to another set B.

A relation from a set A to a set B is a subset R⊆AxB. We often abbreviate the statement (x,y)∈R as xRy.

Definition of relation

A relation on a set A is a subset R ⊆ A × A. We often abbreviate the statement (x, y) ∈ R as xRy.

Define corollary

A result that is an immediate consequence of a theorem or proposition (pg. 88)

Open sentence

A sentence whose true depends on the value of one or more variables (the variables can be a lot things e.g. functions) (pg. 35)

Sequence

A sequence (of real numbers) is a real-valued function defined on the set of natural numbers; that is, a sequence is a function f : N→R. If f (n) = an for each n∈N, then f = {(1, a1), (2, a2), (3, a3),...}.

Convergence of a Sequence

A sequence {a_n} is said to converge to the real number L if for every real number epsilon > 0, there exists a positive integer N such that if n is an integer with n>N, then |a_n - L| < epsilon.

Countable

A set A is called countable if it is either finite, or denumerable.

denumerable

A set A is called denumerable if |A| = |N| to prove: prove a bijective function f: N → A

Denumerable

A set A is called denumerable if |A| = |N|.

Uncountable

A set A is called uncountable if |A|>=|R|

Smaller Cardinality

A set A is said to have smaller cardinality than a set B written as |A|<|B| if there existes a one-to-one function from A to B but no bijective function from A to B.

What does it mean for a set to be uncountable?

A set A is uncountable if it's cardinality is infinite but not countable infinite. That is, A is infinite and there exists no bijection from the naturals to A (pg. 223)

Definition 29.9.

A set S is countably infinite if |S| = |N|. A set is countable if it is either finite or countably infinite.

countable

A set is countable if it is either finite or denumerable.

Zermelo-Fraenkel axioms

A set of axioms for set theory. Includes the well ordered principle and the axiom of foundation: no non-empty set X is allowed to have the property X∩x≠∅ for all its elements x. (pg. 31)

What is a finite set?

A set that has a finite number of elements (pg. 3)

What is an infinite set?

A set that has an infinite number of elements (pg. 3)

What's a contradiction? What are it's values in a truth table?

A statement that can be put in the form C∧~C. It's truth values are all false

Define proposition

A statement that is true but is not as significant as a theorem (pg. 88)

What's a tautology? What are it's values in a truth table?

A statement whose corresponding column in a truth table only contains true.

What is a surjection?

A surjective function (pg. 218)

What is logic?

A systematic way of thinking that allows us to deduce new information from old information and to parse the meanings of sentences (pg. 33)

Define lemma

A theorem whose main purpose is to help prove another theorem or proposition (pg. 88)

Define proof

A written verification of a theorem that shows the theorem is definitely and unequivocally true (pg. 87)

Consider the symbols 0,1,2,3,4,5,6. A) How many such lists are there if repetition is not allowed? B) How many such lists are there if repetition is not allowed and the first three entries must be odd? C) How many such lists are there in which repetition is allowed, and the list must contain at least one repeated number?

A) 7! B) 3*2*1*4*3*2*1=3!4!=144 C) 7^7-7!=818,503 (pg. 71)

If A∩B=∅, what's A-B and B-A?

A-B=A and B-A=B (pg. 18 excercise 2c and 2e)

Union

AUB = {x: xeA or xeB}

Subset

A_c_B if every element of A is also in B.

Compliment

Abar = U - A (basically everything except A)

Theorem 8.9

Addition in Zn, n≥2, is well-defined.

What is the cardinality of the set of even integers E?

Aleph naught.

Union

All of the elements between the two sets

Intersect

All of the elements that both sets have that are the same in both

Cartesian Product

All of the ordered pairs that come from the elements of a set

Tautologies

All of the statements are true

Natural Numbers

All positive integers

Unique

An element belonging to some prescribed set A and possessing a certain property P is unique if it is the only element of A having property P.

What is an injection?

An injective function (pg. 218)

Composite

An integer a for which there exist positive integers b and c, with 1 < b,c < a such that a = bc.

Greatest Common Divisor

An integer c = 0 is a common divisor of two integers a and b if c | a and c | b. Formally, the greatest common divisor of two integers a and b, not both 0, is the greatest positive integer that is a common divisor of a and b. Let a and b be integers that are not both 0. Then gcd(a,b) is the least positive integer that is a linear combination of a and b.

Composite

An integer n is composite if it factors as n=ab where a,b>1 (pg. 90)

definition of even

An integer n is even if n=2a for some integer a∈ℤ. Notice this includes positive and negative numbers (pg. 89)

definition of odd

An integer n is odd if n=2a+1 for some integer a∈ℤ. Notice this includes positive and negative numbers (pg. 89)

Composite

An integer n≥2 that is not prime is called a composite number (or simply composite). An integer n≥2 is composite if and only if there exist integers a and b such that n = ab, where 1 < a < n and 1 < b < n.

Prime

An integer p>1 such that the only positive divisors of p are 1 and p.

Define ordered pair

An ordered pair is a list (x,y) of two things x and y, enclosed in parenthesis and separated by a comma. The order does matter (1,2) ≠ (2,1) (pg. 8)

What is a list?

An ordered sequence of objects, called entries. Order does matter and objects can be repeated. Examples include telephone numbers, zip codes, grocery shopping lists, and more. A list is a group of entries enclosed in parenthesis (pg. 63)

Intersection

AnB = {x: xeA and xeB}

What does intersect represent

And

Overview of contrapositive proof

Another technique used to prove conditional statements (the other is direct proof). It uses the fact that P⇒Q is logically equivalent to ~Q⇒~P (they have the same truth tables). Therefore we can prove P⇒Q by proving ~Q⇒~P.

Real Numbers

Any number that isn't a negative square root

What is a Mersenne prime?

Any prime number of the form 2^n-1 where n∈ℕ (pg. 144)

Transversal

Anything that has at least one element from each partition

Limit (of a sequence)

As we indicated, the number is a measure of how close the terms a_n are required to be to the number L and N indicates a position in the sequence beyond which the required condition is satisfied. If a sequence {an} converges to L, then L is referred to as the limit of {an} and we write lim (n→∞) a_n = L.

Contraditction

Assume P is false, or ~P is true. "By way of contradiction, suppose P and ~Q."

Usually the first step in a set proof

Assume a is an element of the set

Cartesian Product

AxB = {(a,b) : aeA and beB}

cartesian product of sets A and B A={x,y} B={1,2,3}

AxB= {(x,1),(x,2),(x,3),(y,1),(y,2),(y,3)}

disjoint

A∩B={ }

Statements that are so transparent we accept and use them without proof.

A∩B⊆A A⊆A∪B A-B⊆A ((A⊆B)∧(B⊆C))⇒(A⊆C) (X⊆A)⇒(X⊆A∪B) (pg. 135)

What is codomain in "Let f: A -> B be a function"

B is the codomain of f

Prove the fundamental theorem of arithmatic

Be sure to understand the uniqueness part pg. 265

Why are sets so important?

Because all of mathematics can be described with sets (pg. 3)

Why can we make more lists than sets from a set of n things?

Because in lists order matters but in sets it doesn't. E.g (1,2,3)≠(3,2,1) but {1,2,3}={3,2,1} (see pg. 73)

Parity

Both are either even or odd

Minimum Counterexample

By the Well-Ordering Principle, there exists a smallest positive integer n such that P(n) is a false statement. Denote this integer by m. Therefore, P(m) is a false statement, and for any integer k with 1≤k < m, the statement P(k) is true. The integer m is referred to as a minimum counterexample of the statement (6.8). If a proof (by contradiction) of∀n∈N, P(n) can be given using the fact that m is a minimum counterexample, then such a proof is called a proof by minimum counterexample.

What is the cardinality of ℚxℚ?

By theorems 13.4 and 13.5 ℚxℚ is countably infinite/ aleph naught.

C

Complex Numbers

Is composition of functions associative? Commutative?

Composition of functions is associative. So (h◦ g) ◦ f = h◦ (g ◦ f ) Composition of functions is NOT commutative so g◦ f does not necesarily equal f ◦ g (pg. 209)

What are the two categories of existence proofs?

Constructive and non-constructive. Constructive proofs display an explicit example that proves a theorem while non-constructive proofs prove an example exists without actually giving it.

What is the cardinality of an infinite subset of an countably infinite set?

Countably infinite (Theorem 13.8 pg. 230)

surjective reduction

Definition 28.16. Let f : A → B be a function. We define a new function ˆf : A → im(f) by the rule ˆf(a) = f(a) for all a ∈ A. We call ˆf the surjective reduction of f.

Direct, Contrapositive, Contradiction

Direct: Assume P, show Q Contrapositive: Assume -Q, show -P Contradiction: Assume P and -Q. Show F.

Is zero even or odd?

Even because 0=2n where n is an integer, namely zero.

Theorem 34.6

Every integer larger than 1 is divisible by a prime number. pg. 263

Fundamental Theorem of Arithmetic

Every integer n≥2 is either prime or can be expressed as a product of primes; that is, n = p1 p2 · · · p_m, where p1, p2,..., p_m are primes. Furthermore, this factorization is unique except possibly for the order in which the factors occur.

Contradiction

Every possible compound is false

ƒ: A → B

Every x in A must get mapped to some point in B. And no x can get mapped to two places.

True implies False

False

How to prove biconditional statements.

First prove that if p, then Q. Then prove that if Q, then P (pg. 121)

Integers (mod n)

Fix n in N. For a,b in the integers, we say that a is logically equivalent to b (mod n) if n|(a-b).

Define domain, codomain, and range.

For a function f:A→B, the set A is called the DOMAIN of f. (Think of domain as the set of possible "input values" for f.) The set B is called the codomain of f. The range of f is the set {f(a):a∈A}={b: (a,b)∈f}. (Think of the range as the set of all possible "output values" for f. Think of the codomain as a sort of "target" for the outputs.)(pg. 199)

Inverse Relation

For a relation R from a set A to B, the inverse relation R^(-1) from B to A is defined as: R^(-1) = {(b,a}: (a,b)eR}

equivalence class

For an equivalence relation R defined on a set A and for a ∈ A, the set [a] = {x ∈ A : xRa} consisting of all elements in A that are related to a, is called an equivalence class.

Induction

For each positive integer n, let P(n) be a statement. If (1) P(1) is true and (2) the implication: If P(k), then P(k + 1). is true for every positive integer k, then P(n) is true for every positive integer n.

Prove that the cardinality of the powerset of a set, A, is bigger than the cardinality of A.

For finite sets, we know that the cardinality of the power set of A is 2 to the cardinality of A. For infinite sets, see pg. 229-230

Linear combination

For integers a and b, an integer of the form ax + by, where x, y∈Z, is called a linear combination of a and b. Every nonzero integer that divides two integers b and c divides every linear combination of b and c.

Congruent Modulo n

For integers a,b and n>=2, a=_b(mod n) if n|(a-b)

the division algorithm

For positive integers a and b, there exists unique integers q and r such that b= aq + r and 0≤r<a

Converge

Formally then, a sequence {a_n} of real numbers is said to converge to the real number L if for every real number epsilon > 0, there exists a positive integer N such that if n is an integer with n > N, then |a_n−L| < epsilon.

integers modulo n

Furthermore, this equivalence relation results in the n distinct equivalence classes [0], [1],...,[n−1]. We denote the set of these equivalence classes by Zn and refer to this set as the integers modulo n. Thus, Z3 = {[0], [1], [2]} and, in general, Zn = {[0], [1],..., [n−1]}. Hence each element [r] of Zn, where 0≤r < n, is a set that contains infinitely many integers; indeed, as we have noted, [r] consists of all those integers having the remainder r when divided by n.

Relatively Prime

GCD(a,b) = 1

What is a converse?

Given a statement P⇒Q, its converse is Q⇒P. For example, the converse of the statement, "if it is raining, then there are clouds" is "if there are clouds then it is raining." A true statement's converse may or may not be true. (pg. 44)

Compare the cardinality of any set A to the cardinality of its powerset.

Given any set A (finite or infinite), the cardinality of A is less than the cardinality of its powerset (pg. 229)

What does it mean if two integers are congruent modulo n?

Given integers a and b and an n∈ℕ, we say a and b are congruent modulo n if n|(a-b) (if n divides their difference). We express this as a≡b (mod n). If a and b are not congruent modulo n, we write this as a≢ b (mod n). For example, nine and 1 are congruent modulo 4 because the difference of 9 and 1 (8) is divisible by 4. In practical terms, two integers are congruent modulo n if we get the same remainder after dividing a and b by b.

The divisor algorithm

Given integers a and b with b>0, there exist unique integers q and r for which a=qb+r and 0≤r<b

Factorial

Given n is an element of the integers greater than or equal to zero, we define the factorial of n, written n!, as n! = {1 if n=0, n * (n-1)! if n>0}

Given two injective functions, what do we known about their composition? Prove it. Given two surjective functions, what do we know about their composition? Prove it.

Given two injective functions, their composition is injective. Given two surjective functions, their composition is surjective.

Cardinality

How many elements in a set

Can a relation be an equivalence relation on a specific set but not an equivalence relation on different set? Does the equivalence property of a relation depend on the set its defined on?

I'm not sure.

image

If (a,b) ∈ f, then we write b = f(a) and refer to b as ________ of a

Schroder-Bernstein Theorem

If A and B are sets such that |A|≤|B| and |B|≤|A|, the |A|=|B|. prove both one to one, so there exists a bijection A to B, so same cardinality

What is the cardinality of a power set?

If A is a finite set, the |ℙ(A)|=2^|A|. A power set has 2 to the "cardinality of the original set" elements. This stems from fact 1.3 (fact 1.4 pg. 15)

Define power set

If A is a set, the power set of A is another set denoted as ℙ(A) and is defined to be the set of all subsets of A. In symbols, ℙ(A)={X : X⊆A} ALL ELEMENTS OF A POWER SET NEED TO BE SETS (pg. 14)

Proof by contradiction

If P then Q contradiction would be assume P and not Q

Diverge

If a sequence does not converge, it is said to diverge. Consequently, if a sequence {a_n} diverges, then there is no real number L such that lim(n→∞) a_n = L.

Proposition 7.1. Practice proving it.

If a,b∈ℤ, then there exist integers k and l for which gcd(a,b)=ak+bl (pg. 126)

Injective (one-to-one)

If every two distinct elements of A have distinct images in B (aka no two different values of A map to the same value in B)

Fundamental Theorem of Arithmetic

If n in N is greater than 1, then n has a factorization into primes n=p1p2....pr with p1 less than or equal to p2 .... less than or equal to pr and this factorization is unique.

Definition of a factorial

If n is a non-negative integer, then the factorial of n, denoted n!, is the number of non-repetitive lists of length n that can be made from n symbols. Thus 0!=1!=1. If n>1, then n!=n(n-1)(n-2)...(3)(2)(1) (pg. 70)

Fundamental Theorem of arithmetic

If n ∈ N is greater than 1, then n has a factorization into primes n = p1p2 ···pr with p1 ≤ p2 ≤ · · · ≤ pr , and this factorization is unique.

When dealing with expressions dealing with unions and intersections, when MUST we use parenthesis?

If the expression uses both unions AND intersections we MUST use parenthesis. If the expression uses only unions or only intersections, parenthesis are optional (pg 23)

When is f not a well-defined function?

If the rule "Let f: A -> B be a function" produces multiple images for a single element of A

When are two functions f and g equal?

If they have the same domain and codomain, and they are equal as sets of ordered pairs

How many subsets does a finite set, B, have?

If |B|=n, then B must have 2^n subsets because every element can be either inserted or not inserted which means there are two options for every element for a total of 2^n options (fact 1.3 pg. 12)

What is the range of a function

Im(f) = {f(a) : a element A} We cal im(f) the range or the image of f

Trivially True

In Q implies P, it is trivially true if the conclusion (P) is obviously true

Vacuously True

In Q implies P, it is vacuously true if the premise (Q) is never true

How is a list different from a set?

In lists, order does matter and entries can be repeated. Instead of having cardinality, lists have a length (pg. 63)

What is mathematical induction?

Induction is used to prove statements of the form ∀n ∈ N,Sn. It's a means of proving a theorem by showing that if it is true of any particular case, it is true of the next case in a series, and then showing that it is indeed true in one particular case (pg. 154)

Z

Integers {...,-2,-1,0,1,2,...}

I

Irrational Numbers

Recursively defined sequence

Is a sequence of numbers a1, a2, a3, ... where each a0 is given in terms of previous ai (except for the first few).

If A and B are both countably infinite, then AUB ... prove it

Is countably infinite (Theorem 13.6 pg. 227)

What can P(x) mean?

It can represent a statement with the variable x in it (pg. 35)

What does xRy stand for?

It conveys the meaning of the relationship between x and y. Read, "x relates y."

Proof by smallest counterexample

It is a hybrid of induction and proof by contradiction. P(n) is the statement we wish to prove. We prove the base case, say that the statement isn't true for all n, let k be the smallest n such that P(k) is false, and then deduce a contradiction (pg. 165)

What does ~P mean? When is it true? When is it false?

It is not true that P (it's the negation of P) It's true when P is false. It's false when P is true (pg. 40)

What does the Cantor-Bernstein-Schroeder Theorem let us do?

It lets us show that two sets A and B have the same cardinality by finding injections f:A→B and g:B→A. This is useful because injections are often easier to find than bijections.

Explain, "Q is a necessary condition for P"

It means P⇒Q. Given P⇒Q is a true statement (first, third and fourth rows of P⇒Q table), then the only way P is true is if Q is true. Therefore, "Q is a necessary condition for P" (pg. 42)

What does it mean for a an infinite set A to have a smaller cardinality than another infinite set B?

It means that there exists a bijection between A and B but no surjection (pg. 229)

What does P∧Q mean and when is it true? When is it false?

It's a statement that combines two statements P and Q. It's true only when both statements are true at the same time. It's false when one is true and one is false or when both are false (pg. 38)

What does PvQ mean and when is it true? When is it false?

It's a statement that combines two statements P and Q. It's true when either P or Q or both are true. It's false only if both P and Q are false at the same time (pg. 39)

What is a universal set how do we denote it?

It's kind of like the context a set is in. For example the set of prime numbers, P, is in the universe of the set of natural numbers P⊆ℕ. We denote a universal set with the capital letter U (pg. 19)

Is the function f(x)=x^2 (from ℝ to ℝ) injective? Surjective? Bijective?

It's not injective because -2≠2, but f(-2)=f(2). It's not surjective, because if b=-1 (or any negative), then there is no a∈ℝ with f(a)=b. So it's not bijective either (pg. 202)

What is set builder notation?

It's notation used to describe sets that are too big or complex to list between braces. X={expression: rule} where X is all the values of "expression" that are specified by "rule" e.g. E={2n : n∈Z} read, "E is the set of all things of form 2n, such that n is an element of the integers." (see pg. 5 example 1.1)

Open Sentence

It's something like 3x = 12 (true when x=4) It's not really a complete sentence like a statement is, and it's truthfulness depends on other vairables

What is the cardinality of the union of two sets, A and B?

It's the cardinality of A plus the cardinality of B minus the cardinality of their intersection.

What is the cardinality of AxB given sets A and B are finite?

It's the cardinality of A times the cardinality of B (Fact 1.1 pg. 9)

What does ℝxℕ look like?

It's the coordinates in ℝ² that have natural numbers as their Y coordinate (figure 1.2b pg. 9)

What does ℕxℕ look like?

It's the coordinates in ℝ² that have natural numbers as their x and y coordinates (figure 1.2c pg. 9)

What is the cardinality of a set?

It's the number of unique elements a finite set has. AKA it's size. We symbolize cardinality with absolute value bars around the capital letter symbolizing the set (pg. 4)

What is an index set?

It's the set for all the subscripts on our sets (pg. 25)

What does ℝxℝ look like?

It's the set of all points on the 2-dimensional Cartesian plane (figure 1.2a pg. 9)

What is the empty set?

It's the set that has no elements. Symbolized by ∅. ∅={} (pg. 4)

What are Cartesian Powers?

Its the Cartesian product of a set with itself n times (see pg. 10)

Limit of a function

L is the limit of f(x) as x approaches a, written limx->a f(x) = L, if for every real number epsilon > 0, there exists a real number delta > 0 such that for every real number x with 0 < |x-a|<delta, it follows that |f(x) - L|< epsilon.

Greatest Common Divisor

Largest common divisor of a and b when a,b are non-zero integers. Written GCD(a,b).

image

Let A and B be sets and let f : A → B be a function. If S is a subset of A, we define the image of S under f (or just "the image of S") to be f(S) = {f(x) : x ∈ S}. We note that f(S) is a subset of B

Image of Function

Let A and B be sets, and let f : A → B be a function. Define im(f) = {f(a) : a ∈ A}. We call im(f) the range or the image of f. Notice that im(f) ⊆ B.

restriction

Let A and B be sets, and let f : A → B be a function. If S ⊆ A, we can define a new function f|S : S → B by the rule f|S(x) = f(x) for each x ∈ S. We call f|S the restriction of f to S, and read it as "f restricted to S."

Definition 28.9.

Let A and B be sets, and suppose {P1, . . . , Pn} is a partition of A into n parts and {Q1, . . . , Qn} is a partition of B into n parts. Assume that for each i with 1 ≤ i ≤ n, we are given a function fi : Pi → Qi . We may define a function f : A → B by f = [n i=1 fi with the rule f(a) = fi(a) if a ∈ Pi . We call f the function obtained by pasting together the fi

Function

Let A and B be sets. A function f from A to B is a relation from A to B (i.e. a subset of A x B), such that every a ∈ A is a first coordinate of exactly one element of f. In other words ∀a ∈ A, ∃!b ∈ B, (a, b) ∈ f. We call A the domain of f, and B the codomain of f

Relation

Let A and B be sets. A relation from A to B is a subset R subset or equal to A x B. For elements a in A and b in B, we write aRb to mean (a,b) in R. In the case when A=B, we say R is a relation on A.

Relation

Let A and B be two sets. By a relation R from A to B we mean a subset of A × B. That is, R is a set of ordered pairs, where the first coordinate of the pair belongs to A and the second coordinate belongs to B. If (a, b)∈R, then we say that a is related to b by R and write aRb.

Least element

Let A be a nonempty set of real numbers. A number m∈A is called a least element (or a minimum or smallest element) of A if x≥m for every x∈A.

Define the complement of a set A.

Let A be a set with a universal set U. The complement of A is Everything in U that isn't in A (pg. 19)

Equivalence Class

Let A be a set, and let ~ be an equivalence relation on A. For an element a in A, we define the equivalence class of a by [a] = {x in A : a ~ x}. The element a is called a representative of the class [a].

Partition

Let A be a set. A collection P of subsets of A is called a partition of A if P contains non-empty pieces, if every element of A is contained in some element of P, and disjoint pieces.

Russell's paradox

Let A={X: X is a set and X∉X} Is A∈A? For a set X, the equation says that X∈A means that X∉X. For X=A, this then means that A∈A means the same thing as A∉A. So if A∈A is true, then it is false; if A∉A is false, then it is true (pg. 31)

Theorem 8.4

Let P = {Aα:α∈I} be a partition of a nonempty set A. Then there exists an equivalence relation R on A such that P is the set of equivalence classes determined by R, that is, P = {[a] : a∈A}.

The Principle of Mathematical Induction

Let P(n) be an open sentence where the domain of n is N. Suppose that (i) P(1) is true (ii) For all k in N, P(k) implies P(k+1) Then P(n) is true for all n in N.

Quantified Statement

Let P(x) be an open sentence over a domain S. Adding the phrase "For every x∈S" to P(x) produces a statement called a quantified statement. The phrase "for every" is referred to as the universal quantifier and is denoted by the symbol∀. Other ways to express the universal quantifier are "for each" and "for all."

Domain

Let R be a relation from A to B. The domain of R, denoted by dom(R), is the subset of A defined by dom(R) = {a∈A : (a, b)∈R for some b∈B};

Inverse Relation

Let R be a relation from A to B. The inverse relation is the set R −1 = {(b, a) : (a, b) ∈ R} where we reverse each ordered pair in R. It is a relation from B to A

Anti-Symmetric

Let R be a relation on a set A. We say that R is antisymmetric if for all a,b in A, we have (aRb ^ bRa) implies that a=b.

Reflexive

Let R be a relation on a set A. We say that R is reflexive if for all a in A, we have aRa.

Symmetric

Let R be a relation on a set A. We say that R is symmetric if for all a,b in A, we have aRb implies bRa.

Transitive

Let R be a relation on a set A. We say that R is transitive if for all a,b,c in A, we have (aRb ^ bRc) implies aRc.

Equivalence Relation

Let R be a relation on set A. We say that R is an equivalence relation if R is reflexive, symmetric, and transitive.

Theorem 8.3

Let R be an equivalence relation defined on a nonempty set A. Then the set P = {[a] : a∈A} of equivalence classes resulting from R is a partition of A.

Theorem 8.2

Let R be an equivalence relation on a nonempty set A and let a and b be elements of A. Then [a] = [b] if and only if a R b.

Definition 29.1.

Let S and T be sets. We say that S and T have the same cardinality if there exists a bijection f : S → T. If this holds, we write |S| = |T|. If there is no bijection from S to T, we say that they have different cardinalities, and write |S| 6= |T|.

Euclid's Lemma

Let a be an integer > 1. The following are equivalent. (i) a is a prime number (ii) if a|bc then a|b or a|c.

Euclid's Lemma

Let a be an integer and a > 1. If (a) a is a prime number. (b) For b, c as integers, if a divides bc and a does not divide b then a divides c.

Euclid's Lemma

Let a be an integer and a > 1. If (a) a is a prime number. (b) For b, c as integers, if a|bc and a does not divide b then a|c.

Theorem 34.7

Let a be an integer greater than 1. Then a is either prime or is a produce of primes (pg. 264)

Euclid's Lemma

Let a ∈ Z with a > 1. The following are equivalent. (1) a is a prime number. (2) For b,c ∈ Z, if a|bc then a|b or a|c

Euclid's lemma

Let a, b, and c be integers, where a≠0. If a|bc and gcd(a,b) = 1, then a|c Let b and c be integers and p a prime. If p|bc, then either p|b or p|c

Common Divisor

Let a,b,c be integers. If c|a and c|b then c is a common divisor.

Relatively Prime

Let a,b,c in the integers. If GCD(a,b)=1, then we say that a and b are relatively prime.

Composition of two functions

Let f : A → B and g : B → C be functions. We define the composition of g with f, written g ◦ f, to be the new function g ◦ f : A → C given by the relation g ◦ f = {(a, g(f(a))) ∈ A × C : a ∈ A}.

preimage

Let f : A → B be a function, and let T be a subset of B. We define the preimage of T to be the set f −1 (T) = {a ∈ A : f(a) ∈ T}. Note that f −1 (T) is a subset of A.

Injective function

Let f : A → B be a function. We say f is injective if each element of B is the second coordinate of at most one ordered pair in f.

Image

Let f : A → B be a function. We write f(a) = b to mean that (a, b) ∈ f. In other words, f(a) is the second coordinate of the unique ordered pair having a as its first coordinate. When f(a) = b, we say that b is the image of a under the function f.

What is a subjective function?

Let f: A -> B be a function. It is subjective if every element of B appears at least once as the second coordinate of an ordered pair in f.

Pigeonhole Principle

Let m and n be natural numbers with m > n. If m objects are placed in n bins then two (or more) objects must share a bin.

Pigeonhole Principle

Let m and n be natural numbers, with m > n. If m objects are placed in n bins, then two (or more) objects must share a bin.

Explain what is the fundamental theorem of arithmetic is

Let n be an integer greater than 1. Then n has a factorization into primes n=p₁p₂...pk with p₁≤p₂≤...≤pr and this factorization is unique (pg. 264)

The Division Algorithm

Let n,d be elements of the integers with d not equaling zero. Then there are unique integers q,r such that n=qd+r and -1<r<|d|. (r=remainder, q= quotient)

Theorem 8.6

Let n∈Z, where n≥2. Then congruence modulo n (that is, the relation R defined on Z by a R b if a≡b (mod n)) is an equivalence relation on Z.

The Binomial Theorem

Let x,y be variables and let n>-1 be in Z. Then (x+y)^n = summation from k=0 to n of (n choose k) x^(n-k)y^k.

Transversal

Let ~ be an equivalence relation on a set A. A transversal of ~ is a set S subset or equal to A, such that S contains exactly one representative of every equivalence class of ~.

Strong Induction

Like normal induction, but you assume P(l) is true for 1≤l≤k and show that P(k+1) is true.

Why are lists important?

Many real-world phenomena can be described and understood in terms of lists. The world of information technology depends on Bytes, which are lists of length with 0s and 1s.

Diverges to infinity

More formally, a sequence {a_n} diverges to infinity, written lim(n→∞) a_n = ∞, if for every positive number M, there exists a positive integer N such that if n is an integer such that n > N, then a_n > M.

Limit (of a function)

More precisely then, L is the limit of f (x) as x approaches a, written lim(x→a) f (x) = L, if for every real number epsilon > 0, there exists a real number δ > 0 such that for every real number x with 0 < |x−a| < δ, it follows that | f (x)−L| < epsilon.

well-defined

More precisely, addition and multiplication in Zn are well-defined if whenever [a] = [b] and [c] = [d] in Zn, then [a + c] = [b + d] and [ac] = [bd].

Multiplication theorem

Multiplication in Zn, n≥2, is well-defined.

Is ∅={∅}?

No, think of it like boxes. An empty box is not the same thing as an box with an empty box inside of it. ∅ has a cardinality of zero while {∅} has a cardinality of one (pg. 4)

Can only numbers be the elements of a set?

No, we can put letters, coordinates, sets, matrices and more inside of sets (see pg. 4)

Is 1 prime?

No. A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive factors, 1 and the number itself. Natural numbers that have more than two positive factors are called composite. 1 has only one positive factor i.e. no.1 only. Hence 1 is neither prime nor composite

Does xRy mean the same thing as yRx? Explain why with an example.

No. E.g. If the relation is less than, the statement xRy (meaning x is less than y) is not the same thing as yRx (meaning y is less than x).

Is mathematical induction the same thing as inductive reasoning?

No. Inductive reason provides a best guess for a statement while mathematical induction provides certainty (pg. 157)

Does P⇒Q mean the same thing as Q⇒P? In other words does a statement mean the same thing as its converse?

No. See picture (pg. 44)

Irrational Number

Not Rational

Linear Combination

Of two integers a and b is a number of the form ax+by where x,y are integers.

What does union represent?

Or

Define an inverse relation

Out outputs become our inputs and our inputs become our outputs.

Conjunciton

P and Q written PAQ. True only when P and Q are true.

Negation of p implies q

P and not q

Biconditional

P double arrow Q. If and only if Case 1: Assume P...therefore Q Case 2: Assume Q... therefore P

How to express if exactly one of statements P or Q is true

P or Q but not both Either P or Q Exactly one of P or Q (pg. 39)

biconditional

P<=>Q

Biconditional

P<=>Q or (P=>Q)A(Q=>P) True when P and Q have the same truth values

Contrapositive

P=>Q =_ (~Q)=>(~P)

How to find the greatest common divisor (GCD)

Perform long division, and do so until the remain is 0. Whatever you divided by when the remainder was 0 will be the GCD

Theorem 24.2. Congruence modulo n is an equivalence relation on Z.

Proof. (Reflexive): Let a ∈ Z. We have n|0, so n|(a − a). Hence, a ≡ a (mod n), and congruence modulo n is reflexive. (Symmetric): Let a, b ∈ Z, and assume a ≡ b (mod n). Then n|(a − b), so (a−b) = nk for some k ∈ Z. This implies that b−a = n(−k), so n|(b−a), and b ≡ a (mod n). Hence, congruence modulo n is symmetric. (Transitive): Let a,b,c ∈ Z, and assume both a ≡ b (mod n) and b ≡ c (modn). Thena−b=nkandb−c=nlforsomek,l∈Z. Then a − c = (a − b) + (b − c) = nk + nl = n(k + l), so a ≡ c (mod n). Hence, congruence modulo n is transitive.

Prove the following theorem (Theorem 33.12). Let a,b, and c be integers with a not equal to zero. If a divides bc and a and b are relatively prime, then a divides c.

Proof. Assume that a divides bc and GCD(a,b)=1. Since a divides bc we see that bc=ak for some integer k. Also, for some integers x,y we have 1=ax+by. Multiplying the last equation by c, we obtain c=acx+bcy c=acx+aky c=a(cx+ky) So a divides c (pg. 258)

Prove the following theorem: Let a be an integer greater than 1. The following are equivalent. (a) a is a prime number. (b) For integers b and c, if a divides the produce of b and c and a does not divide b, then a divides c.

Proof. Assume that a is a prime number, and that a divides bc and a does not divide b. Then the GCD(a,b) is 1, by theorem 34.4. By Theorem 33.12, we see that a divides c. Now suppose that a is not prime. Then a is composite, so a=bc for some integers b,c with 1<b,c<a. Now a divides bc and a does not divide b and a does not divide c (since b and c are positive and smaller than a. Hence, (b) is false. (pg. 263)

Proof there is infinte prime numbers.

Proof. Let S be any finite nonempty set of prime numbers. Let M = the powe set of all primes in S Then M ≥ 2 since S is nonempty, and any prime is greater than 1. Let N = 1 + M. We note that N > 1, so N is divisible by some prime number p. Suppose, by way of contradiction, that p ∈ S. Then clearly p | M. Since p | N and p | M, we see that p |(N − M), so p | 1. This is a contradiction, since p > 1. Hence, p is a prime that is not in S. Since no finite set of primes can contain all the primes, there must be infinitely many primes.

Let P be a partition of a nonempty set A. There is an equivalence relation ∼ on A such that the equivalence classes of ∼ are precisely the parts of P.

Proof. LetP beapartitionofA. Define the relation ∼ on A by a∼b if a and b are both in a common part of P. We will show that ∼ is an equivalence relation. (Reflexive): Let a ∈ A. Some part of P contains a (by the covering property). Hence, a ∼ a, so ∼ is reflexive. (Symmetric): Let a,b ∈ A and assume a ∼ b. Thus, there is some part X ∈ P, which contains both a and b. Hence, b, a ∈ X, so b ∼ a. Thus, ∼ is symmetric. (Transitive): Let a,b,c∈A and assume a∼b and b∼c. So a,b∈X and b,c∈Y, where X and Y are parts of P. Since both X and Y contain b, we must have X = Y (by the disjoint pieces property), so a, c ∈ X, and hence a ∼ c. Thus, ∼ is transitive, and this finishes the proof that ∼ is an equivalence relation. Finally, we will prove that the equivalence classes of ∼ are exactly the parts of P . For any a ∈ A, there is exactly one part X ∈ P that contains a (by the covering and disjoint pieces conditions). By the definition of ∼, the elements of [a] are exactly the elements of X, so [a] = X. Hence, the equivalence classes of ∼ are parts of P. Conversely, we show that the parts of P are equivalence classes. Given a part X ∈ P , it is non-empty. So fix a ∈ X. By the same argument above we obtain [a] = X, so it is an equivalence class.

Let ∼ be an equivalence relation on a set A, and for x ∈ A, denote the equivalence class of x by [x]. Given x,y ∈ A, the following are equivalent: (1) x ∼ y (the representatives are related). (2) [x] ∩ [y] ̸= ∅ (the classes intersect non-trivially). (3) [x] = [y] (the classes are equal).

Proof. We prove the equivalence by proving three implications: (1)⇒(3), (3)⇒(2), and (2)⇒(1). This will prove that we can move from any one condition to any other. (1)⇒(3): Assume x ∼ y. We wish to show [x] = [y]. We will prove this equality, by showing both containments. First we will show [x] ⊆ [y]. Let z ∈ [x]. We then know x ∼ z. By the symmetric property y ∼ x. By transitivity we obtain y ∼ z. Hence, z ∈ [y], and we have shown [x] ⊆ [y]. Conversely,wenowshow[y]⊆[x]. Letz∈[y]. Theny∼z. Sincex∼y,by transitivity, x ∼ z. Hence z ∈ [x]. Thus, we have shown [y] ⊆ [x]. This finishes the proof that [x] = [y]. (3)⇒(2): Now assume [x] = [y]. By the reflexive property, x ∼ x. Thus x ∈ [x] = [y]. Hence [x] ∩ [y] ̸= ∅ since the intersection contains x. (2)⇒(1): Finally, assume [x] ∩ [y] ̸= ∅. Fix z ∈ [x] ∩ [y]. We then have z ∼ x and z ∼ y. By symmetry and transitivity, x ∼ y.

What is a conditional statement? When is it true and when is it false?

P⇒Q. A way of combining two statements. Given to statements, P and Q, P⇒Q, read "if P then Q" means that Q must be true if P is true. The only way P⇒Q is false is if P is true and Q is false, otherwise P⇒Q is true. Think of it as a promise (pg. 42)

When is P⇔Q true? When is it false?

P⇔Q is true when both P and Q are true or when both P and Q are false. P⇔Q is false when only one of P or Q is true but not both (45)

Come up with a logically equivalent statement for P⇔Q.

P⇔Q=(P∧Q)∨(~P∧~Q) pg. 49

What is the negation of P⇒Q?

P∧~Q

What are statements that have the symbols ∀ and/or ∃ called?

Quantified statements (pg. 51)

Relation

R is a subset of AxB aRb to mean (a, b) is an element of R. then R is a relation on A.

Reflexive

R is reflexive if xRx for every xeA. Or (x,x)eR for every xeA

Equivalence Relation

R is reflexive, symmetric, and transitive.

Symmetric

R is symmetric if whenever xRy, yRX for all x,y e A.

Transitive

R is transitive if whenever xRy and yRz, then xRz for all x,y,z e A. Note: If there are no ordered pairs (x,y) (y,z) then R is transitive vacuously

R is logically equivalent to S

R ≡ S

Relation

R ⊆ AxB aRb to mean (a, b) ∈ R. then R is a relation on A.

Fermat's last theorem

R: For all numbers a,b,c,n∈ℕ with n>2, it is the case that aⁿ+bⁿ≠cⁿ (pg. 36)

Logical Equivelence

R=_S R and S are logically equivalent if R and S always have the same truth values

R

Real Numbers

Congruence modulo n

Recall again that for integers a and b, where a = 0, the integer a is said to divide b, written as a | b, if there exists an integer c such that b = ac. Also, for integers a, b and n≥2, a is said to be congruent to b modulo n, written a≡b (mod n), if n | (a−b).

Partition

Recall that a partition P of a nonempty set S is a collection of nonempty subsets of S with the property that every element of S belongs to exactly one of these subsets; that is, P is a collection of pairwise disjoint, nonempty subsets of S whose union is S.

Goldbach conjecture

S: Every even integer greater than 2 is a sum of two prime numbers (pg. 36)

Cartesian product of n sets

See pg. 9

What is the difference of set A with its complement?

Set A. Because the intersection of A and its complement is the empty set. Therefore we remove no elements from A when finding the difference of A and its complement (pg. 20 exercise 2e)

Power set

Set of all subsets of A, P(A)

Disjoint

Sets A and B have no common element, AnB = empty set

How do we show that an element, a, is an element of a set, S={x: P(x)}?

Show that P(a) is true (pg. 132)

How to prove a∈{x∈S: P(x)}?

Show that a∈S AND P(a) is true (pg. 132)

Binomial theorem

Shows us how to expand any binomial of the form (x+y)^n. The coefficients of each term are nCk where 0≤k≤n.

Compare the cardinalities of the naturals to the reals

Since there is no bijection between the naturals and the reals, their cardinality are not equal.

If A and B are both countably infinite, then... prove it.

So is their Cartesian product, AxB. (Theorem 13.5). Just make a table with the elements of A on one side and the elements of B on the other. Where the columns and rows intersect write the ordered pair correspond to the elements of A and B. Then make a winding path through these ordered pairs.

What are uniqueness proofs? How do we prove them?

Some existence statements have the form "There is a unique x for which P(x). Such a statement asserts that there is exactly one example x=d for which P(d) is true. To prove these, we must come up with an example x=d such that P(d) is true AND we must show that x=d is the only such example (pg. 127)

Statement

Something that is definitely true or definitely false (pg. 34)

Counter Example

Something that verifies that Vx e S, R(x) is false.

What are conjectures?

Statements that have not been shown to be true or false (e.g. the Goldbach conjecture or there are an infinite number of primes) (pg. 147)

Outline for proof by strong induction.

Strong induction works just like regular induction, except that in Step (2) instead of assuming Sk is true and showing this forces Sk+1 to be true, we assume that all the statements S1,S2,...,Sk are true and show this forces Sk+1 to be true. The idea is that if it always happens that the first k dominoes falling makes the (k +1)th domino fall, then all the dominoes must fall.

Formal definition of a function.

Suppose A and B are sets. A function f from A to B (denoted as f: A→B) is a relation f⊆AxB from A to B, satisfying the property that for each a∈A the relation contains exactly one ordered pair of form (a,b). The statement (a,b)∈f is abbreviated f(a)=b. (pg. 197)

Define subset

Suppose A and B are sets. If every element of A is also an element of B, then we say A is a subset of B, denoted A⊆B.

Define countably infinite

Suppose A is a set. Then A is countably infinite if it has the same cardinality as the naturals, that is if there exists a bijection between the naturals and A.

Prove that an infinite subset of a countably infinite set is countably infinite.

Suppose A is an infinite subset of the countably infinite set B. Since B is countably infinite, we can write it's elements in an infinite list by theorem 13.3: b1, b2, b3,.... Then we can also write A's elements in list form by proceeding through the elements of B, in order, and removing the elements that aren't in A. Thus A can be written in list form, making it countably infinite.

What is an equivalence class? How is it denoted?

Suppose R is an equivalence relation on a set A. Given any element a∈A, the equivalence class containing a is the subset {x∈A: xRa} of A consisting of all the elements of A that relate to a. This set is denoted as [a]. Thus the equivalence class containing a is the set [a]={x∈A: xRa}.

Theorem 11.1

Suppose R is an equivalence relation on a set A. Suppose also that a,b∈A. Then [a]=[b] if and only if aRb. (pg. 188)

Theorem 11.2

Suppose R is an equivalence relation on a set A. Then the set {[a]: a∈A} of equivalence classes of R forms a partition of A.

Definition of divisor/multiple

Suppose a and b are integers. We say that a divides b, written a | b, if b=ac for some c∈ℤ. In this case we also say that a is a divisor (factor) of b, and that b is a multiple of a.

How do we show a function f: A→B is surjective?

Suppose b∈B and prove there exists a∈A for which f(a)=b

well-defined function

Suppose that we try to define a function f : A → B by a rule. If the rule produces multiple images for a single element of A, then we say that f is not a well-defined function. If the rule for f produces a single image for every element of A, then we say that f is a well-defined function

What are relations in general?

Symbols that convey relationships. Examples include <,≤,=,|,-,≥,>, ∈ and ⊂, etc

Suppose B is an uncountable set and A is a set. Given that there is a surjective function f : A→B, what can be said about the cardinality of A?

That A is uncountable too.

Reflexive

That is, R is reflexive if (x, x)∈R for every x∈A.

What is the Cartesian product?

The Cartesian product given two sets, A and B, is another set denoted as AxB and is defined as AxB={(a,b) : a∈A and b∈B} notice that the elements of the Cartesian product of two sets are ordered pairs (pg. 8)

Greatest Common Divisor

The GCD of two integers a and b , not both 0, is the greatest positive integer that is a common divisor of a and b.

Example of a non-constructive proof

The Latex homework. (pg. 128)

Basis step and inductive step.

The basis step is the step in mathematical induction in whcih we show the first statement is true. The inductive step is when we show that Sk being true implies the following statement (Sk+1) is true (pg. 156)

How do we denote the cardinality of the naturals?

The cardinality of the natural numbers is ℵ0. That is |N| = ℵ0. This is pronounced "aleph naught"

What is the complement of the set of prime numbers P?

The composite numbers and 1 (pg. 20)

Composition

The composition g o f of f and g is the function from A to C defined by (g o f)(a) = g(f(a)) for all a e A.

Proof by contrapositive

The contrapositive of (If Q then P) would be (Assume If P(not) then Q(not))

Why is 0!=1 and not 0?

The definition factorial says so. Also, if 0!=1 then 1!=0 which is ridiculous. We can show this using the formula n!=n*(n-1)! (pg. 71)

What is the difference of sets A and B (A-B)

The difference of sets A and B is the set of all things that are in A but not in B. Denoted A-B={x:x∈A and x∉B}. Also denoted A-B or A/B

Differentiate between domain, codomain, and range.

The domain is everything that can go into the function. The codomain is everything that could possibly come out of the function. The range is everything that DOES come out of the function.

Power Set

The elements of the power set are the subsets that the power set is of

What is the intersection of a set A with its complement?

The empty set because any set A will share no elements with its complement set (pg. 20 exercise 2c)

Name some important sets

The empty set, the natural numbers, the integers, the rational numbers and the real numbers (pg. 6)

What is the Cartesian product of any set with the empty set ∅?

The empty set. Because |AxB|=|A|*|B| (exercise 1e from 1.2 pg. 10)

What is the difference between ℝx(ℕxℤ) and ℝxℕxℤ?

The first will have elements of the form (a,(b,c)) : a∈ℝ, b∈ℕ, c∈ℤ and the second will be in the form (a,b,c) : a∈ℝ, b∈ℕ, c∈ℤ (pg. 9 exercises 1h,2f,2g,2h pg. 10)

What is the intersection of sets A and B?

The intersection of A and B is the set of all things in BOTH A and B. Denoted A∩B={x:x∈A and x∈B} (pg. 17)

Greatest common divisor

The largest integer that divides both a and b written gcd(a,b). We usually make at least one of a or b not zero (pg. 90)

What is the empty list?

The list with no entries (). It has a length of zero.

Name sets that are countably infinite

The naturals, integers, rationals, etc.

What is a list's length?

The number of entries (including repeats) a list has. So (5,3,5,4,3,3) has a length 6 and (S,O,S) has three (pg. 63)

Compare the Venn diagrams of (A∪B)∩C and A∪(B∩C). Are the parenthesis important?

The parenthesis are essential (pg. 22)

(Strong) Inductive Step

The point in a proof where one uses the inductive hypothesis.

What is disproof?

The process by which we show a statement is false (pg. 146)

Range

The range of R, denoted by range(R), is the subset of B defined by range(R) = {b∈B : (a, b)∈R for some a∈A}.

Partition

The set P = { [a] : a is an element of A } must be: *(Non-empty pieces)* No set in P is empty. *(Covering)* Every element of A is contained in some element of P. *(Disjoint pieces)* Any two distinct elements of P are disjoint.

Describe ℙ(ℝ²)

The set containing any ordered pair or collection of ordered pairs in the 2D Cartesian plane. "In addition to containing every imaginable function and every imaginable black and white image, ℙ(ℝ²) contains the full text of every book ever written." (pg. 16)

The set of rationals, ℚ, is/is not countably infinite. Prove it.

The set of rationals, ℚ, IS countably infinite (Theorem 13.4) (Pg. 224)

Least common multiple

The smallest positive integer that is a multiple of both a and b. Written lcm(a,b) (pg. 90)

Gauss's formula

The sum of the first n natural numbers is n(n+1)/2

What are elements?

The things that make up the set. They are separated by commas inside of braces (pg. 3)

What is the union of sets A and B?

The union of A and B is the set of all things that are in A OR in B. Denoted A∪B={x:x∈A or x∈B} (pg. 17 )

What is the union of set A with its complement?

The universal set that A and its complement are in (pg. 20 exercise 2d)

Use of "and"

The word "and" can be used to combine two or more statements to form a new, more complex statement. The new statement is true if all of the original statements are true (pg. 38)

What is the division algorithm?

Theorem 32.1 (pg. 246)

The division algorithm

There are unique integers q, r such that n=qd+r and 0 <= r < |d|

The division algorithm

There are unique integers q, r such that n=qd+r and 0<= r < abs(d)

continuum hypothesis canter's theorem

There exists no set S such that: |N|<|S|<|R| If f: S→P(S) then f is not surjective

How do we define two sets as having the same cardinality?

There must be a bijective function between the two sets, f:A→B

Compare the cardinalities of the reals and the powerset of the naturals.

They are equal (Theorem 13.11 pg. 236)

What makes two lists equal?

They must have exactly the same entries in exactly the same order. They will thus have the same length. So lists with different lengths are not equal.

What are existentially quantified statements? How do we prove them?

They statements of the form ∃x, R(x). We must simply come up with an example (an x) for which R(x) is true.

Prove or disprove: if a set A is uncountable, then it's cardinality is equal to the cardinality of the reals.

This is false. Let A be the powerset of the reals. Then A is uncountable but it's cardinality is not equal to the cardinality of the reals.

Equation 3.2

This is just stating how to get a part of pascals triangle from the two parts above it. Please see pg. 78

True or false: the cardinality of the naturals is the same as the integers.

This is true because there exists a bijection between them.

Division algorithm lemma. Prove it.

This shows us that the division algorithm produces unique q and r (pg. 246)

Prove or disprove: If A⊆B and A is countably infinite and B is uncountable, then B−A is uncountable.

This statement is true. Suppose to the contrary that B-A is countably infinite. Then Au(B-A)=B is countably infinite (by theorem 13.6). But this a contradiction because B is uncountable (exercise 7 pg. 231)

Cantor-Bernstein-Schroeder Theorem

This tells us that given lAl≤lBl, and lBl≤lAl, then lAl=lBl (pg. 234). In other words if we can construct injections from A to B and from B to A then there is a bijection from A to B

When do we use the Euclidian Algorithm?

To find the GCD 57 = 1*39 + 18 39 = 2*18 + 3 18 = 6*3 + 0 GCD is 3

Proof by contradiction

To prove "if P then Q" by contradiction, begin by assuming P and not Q, then you are finished when you find a contradiction Take out "if...then" and put "and" (and negate the second)

Outline for proof by contradiction

To prove a statement P is true, we assume it's not true and then show that this leads to something absurd (like 1 is even or something pg. 112)

False implies True

True

What happens if we start with a statement and add "it is not true that" to the beginning of it?

True statements become false and false statements become true (pg. 40)

Formal definition of equality of functions

Two function f: A→B and g: A→D are equal if f(x)=g(x) for every x∈A. Observe that f and g can have different codomains and still be equal!!! (pg. 200)

equal functions

Two functions f and g are equal if they have the same domain and codomain, and they are equal as sets of ordered pairs.

Relatively Prime

Two integers a and b, both not 0, are called relatively prime if gcd(a,b) = 1.

Relatively Prime

Two integers a and b, not both 0, are called relatively prime if gcd(a, b) = 1. Let a and b be integers, not both 0. Then gcd(a, b) = 1 if and only if there exist integers s and t such that 1 = as + bt.

Definition of parity

Two integers have the same parity if they are both even or they are both odd. Otherwise they have opposite parity. E.g. 5 and -17 have the same parity (pg. 89)

Logically equivalent

Two or more statements are logically equivalent if their truth tables match up line-for line in a truth table (pg. 49)

How do we symbolically say an element is part of a set?

Using the ∈ symbol. e.g. If S={a,b,c,d,e...} a∈S read "a is an element of S" or, "a is in S" or, "a in S." We could also say a,b,c∈S (pg. 3)

How do we represent sets?

Usually with uppercase letters e.g. A={1,3,5,7,9...}

Universal Quantifier

V "for every" Vx e S, P(x)

How do we disprove a statement P with contradiction?

We assume P is true and deduce a contradiction (pg. 152)

How do we show a function f: A→B is injective?

We can do so directly: suppose x,y∈A and x≠y. .... Therefore f(x)≠f(y). We can also take the contrapositive approach: suppose x,y∈A and f(x)=f(y). .... Therefore x=y (pg. 203)

If we can show that there exists an injection from (infinite sets) A to B, what can we say about the relative size of their cardinalities?

We can say that the cardinality of A is less than or equal to the cardinality of B (pg. 229)

Use of "or"

We can use the word "or" to combine two or more statements to form a new, more complex statement. The new statement is true if at least one of the original statements is true (pg. 39)

How do we disprove a conditional statement, P⇒Q?

We find an example for which P is true but Q is false (pg. 149)

Outline for proof by induction. Proposition: S1,S2,S3,S4,...Sn are true (∀n ∈ N,Sn).

We first prove S1 is true. Then we prove that given any integer k greater than one, the statement Sk⇒Sk+1. It then follows by induction that every Sn is true (pg. 156)

Go over example 3.2 part d.

We have to subtract the total number of lists without an E (allowing repetition) from the total number of lists (allowing repetition). If you do it the wrong way, you count some of the lists twice. pg. 66

Prove the following theorem: Let p be a prime number, and let a be an integer. Then GCD(p,a) is p if p divides a or is 1 if p does not divide a.

We know that GCD(p,a) must be a positive divisor of p, so it must be 1 or p (the only two divisors of p). If p divides a then p is clearly the largest common divisor; similarly if p does not divide a, then 1 is the largest common divisor (pg. 263)

How do we prove that a set, A, is a subset of another set, B?

We must prove that if a∈A, then a∈B for all a∈A (pg. 133). We can do this directly (suppose a∈A...A⊆B), contrapositively (Suppose a∉B...A⊆B), or using a contradiction (Suppose (a∈A)∧(a∉B)⇒(C∧~C)

How do we prove a function f is NOT surjective?

We must prove the negation of the statement ∀b∈B, ∃a∈A, f(a)=b, that is, we must prove ∃b∈B,∀a∈A, f(a)≠b (pg. 203)

How do we disprove existence statements?

We prove that statement is false for all x. In other words, we prove its negation (pg. 150)

How do we prove a statement is false?

We prove the negation of the statement is true using a direct, contrapositive, or contradiction proof (pg. 148)

Antisymmetric

We say that R is antisymmetric if for all a,b element A, we have ((aRb) and (bRa)) them a = b (If it works for both in both ways, then they are the same number.

Reflexive

We say that R is relexive if for all a element A, we have aRa (If you put in the same element, the equation still applies)

Symmetric

We say that R is symmetric if for all a,b element A, we have aRb implies bRa (You put the values in both positions, and you see if the equation still holds)

Transitive

We say that R is transitive if for all a,b,c element of A, we have ((aRb) and (bRc)) implies (aRc) (so if it works for a and b, and for b and c, then it must for work a and c to be transitive

Greatest element

We say x is the greatest element of S if for al y element S, we have x is greater than or equal to y

Least element

We say x is the least element of S if for al y element S, we have x is less than or equal to y

How do we prove two sets, A and B, are equal?

We show that A⊆B and B⊆A.

How do we disprove universally quantified statements?

We simply find an example that makes it false.

When dealing with counting sets or lists, when do we use permutation and when do we use combination?

We use combinations for sets because sets with the same elements in different order are equivalent. We use permutations for lists because lists with the same entries in different order are not equivalent.

Prove that If U⊆A, and U is uncountable, then A is uncountable.

We use contradiction and theorem 13.8 (pg. 231)

Outline for proving a conditional statement with Contradiction

We want to prove if P, then Q. We start with "suppose P and ~Q" and end up with C∧~C (pg. 115)

What is a vacuous statement?

When given a conditional statement P⇒Q, if it can be shown that P is always false, then we have a vacuous statement.

What is a trivial statement?

When given a conditional statement P⇒Q, if it can be shown that Q is always true, then we have a trivial statement.

When is nCk=0?

When k<0 or k>n (pg. 74)

Well-defined

When we have a definition that appears to depend on arbitrary choices, but for which the arbitrary choices can be shown to make no difference in the definition, then the object being defined is well-defined. (i.e. addition on Zn is well-defined).

Integers

Whole numbers

Prove that there are inifinte many primes

Write P as 2,3,5,7,11,13,... by Theorem 13.3, this is countable infinite.

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(f (X)) is a subset of what?

Y

Is the empty set, ∅, finite?

Yes and it has a cardinality of zero (google)

Do the intervals (0,1) and (0,∞) have the same cardinality?

Yes, because there is a bijective function between these two sets

Can a set be a subset of itself?

Yes, every element in A is in A. e.g. {2,3,7}⊆{2,3,7} all sets are subsets of themselves (example 1.2 #3 pg. 11)

Strong Induction

You have a stronger basis in proving p(k+1) is true because of multiple base cases that are used.

find the equivalence classes R={(1,1), (1,2), (2,2), (3,3), (3,1), (3,2)}

[1]= {1,2} [2]= {2} [3]= {1,2,3}

Equivalence class

[a] = {x is an element of A : a~x }

Equivalence class

[a] = {x ∈ A : x ~ a }

Equivalence Class

[a] = {xeA: xRa} consisting of all elements of a in A is an equivalence class. Basically the equivalence class of a, [a] is the set of all the numbers or variables that a is paired up with in the relation.

Intergers mod n

a congruent b (mod n) nk = a - b

open sentence

a declarative sentence that contains one or more variables ex. 3x=12 domain of x=4

statement

a declarative sentence that is true or false ex. The integer 3 is odd.

Divides

a divides b if there is come integer c such that b = ac. Write: a|b.

Theorem 18.13 If (a divides bc) and gcd(a, b) = 1 , then

a divides c

sequence

a function f: N→R

in order for a relation to be a function...

a relation, R, from A to B is a function if every element of A appears exactly once as the first entry in a ordered pair of R

uncountable

a set that is not countable

Integers mod n

a ≡ b (mod n) nk = a - b

Theorem 18.15 If (a divides c) and (b divides c) and gcd(a, b) = 1 , then

ab divides c

Theorem 18.15 If (a divides c) and (b divides c) and gcd(a, b) = 1 , then

ab|c

Given n countably infinite sets A1,A2,A3,....,An with n≥2, their cartesian products have a cardinality of ... prove it.

aleph naught. In other words the cartesian product of all of them is countably infinite (Corollary 13.1 pg. 227)

contradiction

all possible combos are false ex. P and not P

tautology

all possible combos are true ex. P or not P

Define definition

an exact, unambiguous explanation of the meaning of a mathematical word or phrase (pg. 87)

common divisor

an integer c≠0 is a ______ of two integers a and b if c|a and c|b.

Prime

an integer p > 1 such that the only positive divisors of p are 1 and p.

relation of set A to B

any subset of AxB

Start a direct proof (P implies Q)

assume P is true show Q is true

Start a contrapositive proof (P implies Q)

assume Q is false show P is false

Theorem 18.13 If (a divides bc) and gcd(a, b) = 1 , then

a|c

Reflexive

a~a

Symmetric

a~b and b~a

Existential Quantifier

backwards E "There exists" Ex e S, P(x)

two sets having the same cardinality

both are empty or there is a bijection from one to the other

Form intervals given two numbers a,b∈R : a<b

closed interval [a,b], half open interval (a,b] or [a,b), open interval (a,b), infinite interval (a,oo] (pg. 6)

The cardinality of the cross product of the integers and rationals is ....

countably infinite because the cardinality of both sets is countably infiinite, so their cross product is too by theorem 13.5

countably infinite

denumerable sets

Domain

dom R = {aeA: (a,b)eR for some b e B} (basically the "a" values of the relation)

Let R be a relation from A to B. R= {(a,1), (b,3), (b,4), (d,4)} What is domR and ranR?

domR= {a,b,d} ranR= {1,3,4}

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f (W ∪ X) equals what?

f (W ∪ X) = f (W)∪ f (X)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f (W ∩ X) is a subset of what?

f (W)∩ f (X)

Composite

for positive integers b and c, with 1 < b and c < a such that a = bc.

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(Y ∩ Z) equals what?

f−1(Y)∩ f−1(Z)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. f−1(Y ∪ Z) equals what?

f−1(Y)∪f−1(Z)

Given f : A → B is a function. Let W, X ⊆ A, and Y,Z ⊆ B. X is a subset of what?

f−1(f (X))

Define the composition of two functions

g∘f=g(f(x)) f∘g=f(g(x))

Antisymmetric

if (a~b and b~a) then a=b

Transitive

if (a~b and b~c) then a~c

A set is countably infinite if...

if and only if it's elements can be arranged in an infinite list (Theorem 13.3 pg. 223)

Common divisor of a and b

is a nonzero integer c such that c divides a and c divides b.

Common divisor of a and b

is a nonzero integer c such that c|a and c|b.

Linear Combination of a and b

is a number of the form ax + by where x, y are integers.

greatest common divisor of a and b

is the greatest positive integer that is a common divisor of a and b

GCD(a, b)

is the largest common divisor of a and b.

True/false 1. (∀x ∈ A)(∃y ∈ A)(x+y is even) 2. (∃y ∈ A)(∀x ∈ A)(x+y is even) don't know A

know (2) is true, then (1) is true know (1) is true, then you don't know if (2) is true.

The Division Algorithm

n = qd + r where d is not 0, and r is greater than or equal to 0 and less than the magnitude of d

Binomial Coefficient

n choose k = { n!/((k!)(n-k)!) if 0<=k<=n 0, otherwise }

Binomial Coefficient

n choose k = { n!/((k!)(n-k)!) if 0≤k≤n 0, otherwise }

Factorial

n! = { 1, if n=0 and (n)(n-1)! , if n>0}

Factorial

n! = { 1, if n=0 and (n)(n-1)...2,1 , if n>0}

What is nC1 where n is a natural number? What is it equal to?

n. It's equal to nC(n-1)

How many trailing zeros are in the expansion of n!?

n/5+n/5^2+...n/5^k where k must be chosen s.t. 5^(k+1)>n. Please see http://www.purplemath.com/modules/factzero.htm

What is the smallest n for which n! has more than 10 digits?

n=14

Interpret the meaning of nCk.

nCk is the number of subsets of cardinality k that can be made from a set of cardinality n. In other words, ... (pg. 75)

How many subsets can be made by choosing k elements from a set of n elements?

nCk read, "n choose k".

How many non-repetitive lists of length k can be made if the entries are chosen from a set with a cardinality of n?

nPk

bijective

one-to-one and onto

Show that P is logically equivalent to (~P)⇒(C∧~C)

pg. 112

Definition of common divisor

pg. 120

Go over example 1.3 and make sure you understand all of the statements

pg. 13

Be sure to understand the following: X∪Y=Y∪X and X∩Y=Y∩X, but in general X-Y≠Y-X

pg. 17

Definition of reflexive, symmetric, and transitiive relations

pg. 179

Considering a graph of a relation, how can you tell if a relation is reflexive? Symmetric? Transitive?

pg. 181

Define the integers modulo n.

pg. 192

Demonstrate that the elements of the integers mod n obey the commutative and distributitive laws.

pg. 192

Summarize the ideas presented on pg. 193. Can you prove it?

pg. 193

Prove that composition of functions is associative

pg. 209

Compare the Venn diagrams of A∪B∪C and A∩B∩C

pg. 21

Define image and preimage

pg. 215

Show that equality of cardinalities is an equivalence relation on sets. In other words, show that equality of cardinalities is reflexive, symmetric, and transitive.

pg. 221

Union and intersection of n FINITE sets using notation similar to sigma notation

pg. 24

Theorem 32.4. Prove that given nonzero integers a and b, and a divides b, then the magnitude of a is less than the magnitude of b

pg. 247

Prove the following theorem: let a be an integer greater than 1. If a is composite then there are positive integers b and c, with 1<b,c<a such that a=bc.

pg. 262

Distinguish between correct logic and correct information

pg. 33

Reread last part of section 2.3

pg. 43-44

Go over exercises for 2.3

pg. 44

Other ways to say P if and only if Q

pg. 46

Associative laws

pg. 50

Commutative laws

pg. 50

Contrapositive law

pg. 50

DeMorgan's laws

pg. 50

Distributive laws

pg. 50

Pascal's triangle

pg. 78

Show that the nth row of Pascal's triangle lists the coefficients of (x+y)^n

pg. 79

What is the addition principle?

pg. 82

Prime

prime is an integer p≥2 whose only positive integer divisors are 1 and p.

Range

ran R = {beB: (a,b)eR for some a e A} (basically the "b" values of the relation)

range of f: A→B

ran f = {b ∈ B : bis an image under f of some element of A} = {f(x) : x ∈ A} consists of the second coordinates of the elements of f

Summarize the four different injective/surjective combinations that a function may posses.

see the pic

compound statement

statement composed of one or more statements ex. (P v Q) ∧ (P v R)

What is the fundamental theorem of arithmetic?

states that every natural number greater than 1 has a unique factorization into prime numbers (pg. 118)

sequence of partial sums

s₁=a₁, s₂=a₁+a₂, ...

logical connectives

the symbols ∼,v,∧,=>,<=>

or

v

Let R be a relation from A to B domR=

{a∈A: (a,b)∈R for some b∈B}

Let R be a relation from A to B ranR=

{b∈B: (a,b)∈R for some a∈A}

ℙ({∅})=?

{∅,{∅}} (pg. 15)

ℙ(∅)=?

{∅} (a set containing all the subsets of the empty set, namely the empty set pg. 15)

Triangle Inequality:

|x + y|=<|x|+|y| also note |x| = {x if x>=0 and -x if x<0

What's the negation of ∀x∈S, P(x)?

~(∀x∈S, P(x))=∃x∈S, ~P(x).

What the negation of an existence statement, ∃x∈S, P(x)?

~(∃x∈S, P(x))=∀x∈S, ~P(x) (pg. 150)

Rational numbers

ℚ={x : x=m/n, where m,n∈ℤ and n≠0) The set of all numbers that can be expressed as a fraction of two integers (pg. 6)

Real numbers

ℝ. It's the set of all real numbers on the number line or negative infinity to infinity (pg. 6)

Let A={(x,x^2):x∈ℝ} What does the complement of A look like?

ℝ²-A (pg. 20)

What is the set of divisors of 0?

ℤ because zero can be divided by any integer (pg. 90)

Integers

ℤ={...,-2,-1,0,1,2,...} positive and negative whole numbers (pg. 6)

What does the symbol ⇔ mean?

⇔ expresses the meaning of (P⇒Q)∧(Q⇒P) and It's read, "P if and only if Q." (pg. 45)

What does the symbol ∀ mean?

∀ means "for all" or "for every." It's the universal quantifier. A statement with ∀ in it is universally quantified (pg. 51)

What does the symbol ∃ mean?

∃ means "there exists a" or "there is a". It's the existential qualifier. A statement with a ∃ in it is existentially quantified (pg. 51)

If A⊆B, what's A-B?

∅ b/c all elements in A are in B so there's nothing in A that's not in B so A-B is the empty set (pg. 18 exercise 2h)

Difference between ∧,v, and ∼

∧=and, v=or, and ~="it is not true that"/negation (section 2.1)

Describe the relation ≡ (mod n) on the integers. Is this relation reflexive? Symmetric? Transitive? Prove it.

≡ (mod n) is reflexive, symmetric, and transitive making it an equivalence relation (pg. 182)