MATH 290
Example of a constructive proof
(pg. 128)
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)
What is the multiplication principle?
A fast way of counting (pg. 65)
Theorem 18.12
A fast way to compute the GCD.
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)
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)
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
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)
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)
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)
Goldbach conjecture
S: Every even integer greater than 2 is a sum of two prime numbers (pg. 36)
Compare the Venn diagrams of (A∪B)∩C and A∪(B∩C). Are the parenthesis important?
The parenthesis are essential (pg. 22)
What is disproof?
The process by which we show a statement is false (pg. 146)
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)
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)
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)
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
Natural numbers
ℕ={1,2,3,4,5...} (positive whole numbers pg. 6) Sometimes zero is included in the natural numbers
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)
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)
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)
What is an ordered triple?
(x,y,z) (pg. 9)
What is a bijection?
A bijective function. I.e. a function that is both injective and surjective
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.
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.
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)
Define a bijective function
A function is bijective if it is both injective and surjective (pg. 201).
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 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
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)
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)
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
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)
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)
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)
Define proof
A written verification of a theorem that shows the theorem is definitely and unequivocally true (pg. 87)
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)
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)
What is the cardinality of the set of even integers E?
Aleph naught.
What is an injection?
An injective function (pg. 218)
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)
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)
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.
What is a Mersenne prime?
Any prime number of the form 2^n-1 where n∈ℕ (pg. 144)
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)
What is the cardinality of ℚxℚ?
By theorems 13.4 and 13.5 ℚxℚ is countably infinite/ aleph naught.
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)
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
How to prove biconditional statements.
First prove that if p, then Q. Then prove that if Q, then P (pg. 121)
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
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.
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.
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)
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)
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)
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)
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 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)
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)
What is an identity function?
It's outputs are it's inputs. e.g. y=x
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)
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)
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 34.7
Let a be an integer greater than 1. Then a is either prime or is a produce of primes (pg. 264)
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)
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.
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)
Define an inverse relation
Out outputs become our inputs and our inputs become our outputs.
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)
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)
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)
Come up with a logically equivalent statement for P⇒Q
P⇒Q=(~Q)⇒(~P) pg. 49
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)
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)
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)
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)
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)
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.
What is the complement of the set of prime numbers P?
The composite numbers and 1 (pg. 20)
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
What are relations in general?
Symbols that convey relationships. Examples include <,≤,=,|,-,≥,>, ∈ and ⊂, etc
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)
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.
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)
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"
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)
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.
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)
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)
How do we define two sets as having the same cardinality?
There must be a bijective function between the two sets, f:A→B
Ways to express P⇒Q
These all can be replaced with, "If P, then Q" pg. 43
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
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)
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)
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...}
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)
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 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.
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)
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)
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)
Define definition
an exact, unambiguous explanation of the meaning of a mathematical word or phrase (pg. 87)
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)
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)
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))
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
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
Union and intersection of an INFINITE number of sets
pg. 25
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
Do questions 1.8:9,13
pg. 28
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
Summarize the four different injective/surjective combinations that a function may posses.
see the pic
What is the fundamental theorem of arithmetic?
states that every natural number greater than 1 has a unique factorization into prime numbers (pg. 118)
ℙ({∅})=?
{∅,{∅}} (pg. 15)
ℙ(∅)=?
{∅} (a set containing all the subsets of the empty set, namely the empty set pg. 15)
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)
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)
