Abstract Math Final

subset

If A and B are sets, then we say that A is a ____________ of B, written A ⊆ B , provided that every element of A is also an element of B

elements

If A is a set and x is an ______________ of A, we write x ∈ A. Otherwise, we write x ∈/ A

proper subset

If A ⊆ B, then A is called a proper subset provided that A , B. In this case, we may write A ⊂ B or A ⊂ but not equal to B .

power set

If S is a set, then the power set of S is the set of subsets of S. The ___________________ of S is denoted P(S)

corollary of bezout's lemma

If a and b are relatively prime, then there exist integers s and t such that 1 = as + bt.

existence of additive identity

There is an element 0 ∈ Z such that x+0 = x, where x is any integer

how to check for equivalence relation

proof that a relation is reflexive, symmetric, and transitive

additive law for <

if x < y then x + n < y + n, where x,y,n are integers

universal conditional statement

"∀x, if P (x) then Q(x)"

union of an entire collection of sets

(big U) α∈∆) Aα = {x | x ∈ Aα for some α ∈ ∆}.

intersection of an entire collection of sets

(big n) α∈∆) Aα = {x | x ∈ Aα for all α ∈ ∆}

multiplicative law for <

. If 0 < n and x < y, then nx < ny, where x,y,n are integers

equivalence relation

. Let ∼ be a relation on a set A. Then ∼ is called an _______________________ if and only if ∼ is reflexive, symmetric, and transitive, proof for each one

intersection of entire collection example

4.37, homework 16, for each r in rational umbers, let N, be the set containing all real numbers except r, {x such that x in N sub r for all r in Q} = all real numbers (R) - rational numbers (Q)

union of entire collection example

4.37, homework 16, for each r in rational umbers, let N, be the set containing all real numbers except r, {x such that x in N sub r for some r in Q} = (negative infinity, infinity)

set builder notation example (4.4)

4.4, homework 11, A = { x in N such that x = 3k for some k in N}, N is natural numbers, answer: the set A includes the natural numbers x, where 3 divides x

suppose ~ is an equivalence relation on a set A and let a, b be an element of A. Then [a] = [b] if and only if a ~ b

6.30, homework 24, look at homework, use reflexive property and transitive property for set equality

proof that a positive integer n greater than 1 can be written as a product of primes

7.1, homework 21, induction, P(n): n can be written as a product of primes for all integers greater than 1, base: P(2), inductive step K >= 2 assume P(i) for all 2<=i<=k, WWTS P(k+1) true, consider k+1 is prime and not prime

proof for gcd(a, b) = gcd(b, r) where a = bq + r

7.10, homework 24, so d=d' by showing d <= d' (7.8) and and d' <= d (7.9)

proof of Bezout's corollary

7.15, homework 25, if a and b are relatively prime, then their greatest common divisor would be 1. Therefore, following Bezout's lemma, there exists integers s and t where 1= as + bt (only needed to do short explanation_

proof of Euclid's lemma

7.16, homework 25, assume otherwise. That is, assume that p does not divide a, so that p and a are relatively prime." Next, apply Lemma 7.15 to p and a and then multiply the resulting equation by b. Try to conclude that p divides b, start with 1= ps = at, multiply bot sides by b, p divides ab definition into equation, show p divides b

proof where d is a common divisor of a and b so must also divide r (a = bq + r)

7.8, homework 23, d divides a and d divides b definition, plug that into a = bq + r, show r is divisible by d

proof where d' is a common divisor of b and r so must also divide a (a = bq + r)

7.9, homework 23, d' divides b and d' divides r definition, plug that into a = bq + r, show a is divisible by d'

pairwise disjoint

A collection of sets {Aα}α∈∆ is _______________ if Aα ∩ Aβ = ∅ for α , β

partition

A collection Ω of subsets of a set A is said to be a _______________ of A if the elements of Ω satisfy: (a) Each X ∈ Ω is nonempty, (b) Given X,Y ∈ Ω, either X = Y or X ∩Y = ∅, and (c) union of collection of set X, X∈Ω, X = A In other words, a ____________________, ω, of a set A is a collection of subsets of A that are pairwise disjoint and their union is all of A

contradiction

A compound statement that is always false

tautology

A compound statement that is always true

rational

A number that can be made by dividing two integers (an integer is a number with no fractional part)

predicate

A sentence with a free variable is called this

corollary

A theorem that follows almost immediately from another theorem

converse

A → B is B → A

contrapositive

A → B is ¬B → ¬A

n-tuple

An __________________ is an object of the form (x1 , x2 ,..., xn). Each xi is referred to as the ith component. Two n-tuples (x1 , x2 ,..., xn) and (y1 ,y2 ,...,yn) are equal if and only if xi = yi for all 1 ≤ i ≤ n.

ordered pair

An ___________________ is an object of the form (x,y). We say that two __________________ (x,y) and (a,b) are equal, and write (x,y) = (a,b), if and only if x = a and y = b

divides

An integer n divides the integer m, written n|m, if and only if there exists k ∈ Z such that m = nk. Equivalently, we may also say that m is divisible by n

odd

An integer n is this if n = 2k + 1 for some k ∈ Z

even

An integer n is this if n = 2k for some k ∈ Z

composite

An integer, n, is called ___________ if and only if n > 1 and we can write n = rs for some integers r and s such that 1 < r < n and 1 < s < n.

prime

An integer, n, is called _____________ if and only if n > 1 and for all positive integers r and s, if n = rs, then either r or s equals n

Euclid's lemma

Assume that p is prime. If p divides ab, where a and b are positive integers, then either p divides a or p divides b

Fundamental Theorem of Arithmetic

Every positive integer greater than 1 can be expressed uniquely (up to the order in which they appear) as the product of one or more primes, The Fundamental Theorem of Arithmetic is one of the many reasons why 1 is not considered a prime number. If 1 were prime, prime factorizations would not be unique.

interval notation

For a,b ∈ R with a < b, we define the following.(a) (a,b) = {x ∈ R | a < x < b}(b) (a,∞) = {x ∈ R | a < x}(c) (−∞,b) = {x ∈ R | x < b}(d) [a,b] = {x ∈ R | a ≤ x ≤ b}

law of trichotomy

For all integers x and y, exactly one of the following is true: x > y or x < y or x = y

closure of addition

For all integers x and y, it is true that x + y ∈ Z

commutativity of addition

For all integers x and y, it is true that x+y = y+x.

closure of multiplication

For all integers x and y, it is true that xy ∈ Z.

commutativity of multiplication

For all integers x and y, xy = yx

associatively of addition

For all integers x, y, and z, it is true that (x +y) +z = x + (y +z)

distributive property

For all integers x,y, and z, it is true that x(y + z) = xy + xz

associativity of multiplication

For all integers x,y, and z, it is true that x(yz) = (xy)z

Bezout's lemma

For any nonzero integers a and b, there exist integers s and t such that gcd(a,b) = as + bt

additive inverse property

For each x ∈ Z, there exists −x ∈ Z such that x + (−x) = 0

relatively prime

If gcd(m,n) = 1, we say that m and n are ___________________

disjoint

If two sets A and B have the property that A ∩B = ∅, then we say that A and B are ______________ sets

law of transitivity

If x < y and y < z, then x < z, where x,y, z are integers

how to find cartesian product of two sets

Let A = {a,b, c} and B = {1,2}. Then A × B = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)}

DeMorgan's laws for sets

Let A and B be sets. Then (a) (A ∪B) c = A c ∩B c (b) (A ∩B) c = A c ∪B c .

strong induction

Let P (n) be a property defined for integers n, and suppose that a and b are fixed integers, with a ≤ b. Suppose that the following two statements are true1. Base step: P (a),P (a + 1),P (a + 2),...,P (b) are all true2. Inductive step: For any integer k ≥ b, if P (i) is true for all integers i from the smallest value a all the way up to and including k, then P (k + 1) is true.Then, the statement ∀ integers n ≥ a, P (n) is true

ordinary induction

Let P (n) be a property that is defined for integers n, and let a be a fixed integer. Suppose the following two statements are true: 1. P (a) is true2. For all integers k ≥ a, if P (k) is true, then P (k + 1) is true. Then, the statement for all integers n ≥ a,P (n) is true

universal statement

Let P (x) be a predicate and let D be the universe of discourse for x, "∀x ∈ D,P (x)". It is defined to be true if and only if P (x) is true for every value of x in D. It is defined to be false if and only if there is at least one x in D such that P (x) is false. A value of x for which P (x) is false is called a counterexample to the _____________________.

existence of multiplicative identity

There exists an element, 1 ∈ Z such that for all integers, x, x·1 = x

existential statement

Let P (x) be a predicate and let D be the universe of discourse for x, "∃x ∈ D,P (x)". It is defined to be true if and only if there is at least one x in D for which P (x) is true. It is defined to be false if and only ifP (x) is false for every x in D.

relation

Let X and Y be sets. A _____________, R, from X to Y is a subset of X × Y . A _______________ on X is a subset of X × X. Using usual set notation, we may write (x,y) ∈ R if the ordered pair (x,y) is in the _______________, R. If (x,y) ∈ R, we may also equivalently write xRy or x ∼ y and say that x is ______________ to y under the ____________ R

Division Algorithm

Let m and n be integers with n > 0. Then there exist unique integers q and r with the property that m = nq + r with 0 ≤ r < n

greatest common divisor

Let m,n ∈ Z such that at least one of m or n is nonzero. The ___________________________ of m and n, denoted gcd(m,n), is the largest positive integer that is a factor of both m and n

proof for the fundamental theorem of arithmetic

Let n be a natural number greater than 1. By Lemma 7.1 (product of primes), we know that n can be expressed as a product of primes. All that remains is to prove that this product is unique (up to the order in which they appear). For sake of a contradiction, suppose p1p2 ···pk and q1q2 ···ql both prime factorizations of n. Your goal is to prove that k = l and that each pi is equal to some qj . Make repeated use of Euclid's Lemma

Z sub n for a given integer n

Let n ∈ N. The equivalence classes of the equivalence relation ≡ (mod n) are [0],[1],[2],...,[n − 1]. The integers modulo n is the set Zn = © [0],[1],[2],...,[n − 1]ª . Elements of Zn can be added by the rule [a]+[b] = [a+ b] and multiplied by the rule [a]·[b] = [ab].

equivalence calss

Let ∼ be an equivalence relation on a set A and let x ∈ A. Then we define the ___________________ of x to be the set of all elements in A that are related to x under ∼. We denote the __________________ of x by [x] and write its definition in symbols as [x] = {y ∈ A | y ∼ x}. Also, we define Ω∼ = {[x] | x ∈ A} to be the set of all __________________ of ∼ in A. Notice that Ω∼ is a set of sets. In particular, each element of Ω∼ is a subset of A (equivalently, an element of P(A))

proof by strong induction skeleton proof

Proof. We proceed by induction.(i) Base step: [Verify that P (a) is true. Depending on the statement, you may also need to verify that P (i) is true for other specific values of i.](ii) Inductive step: Let k ≥ a ∈ Z + . Suppose P (i) is true for all i ≤ k. [Do something to derive that P (k + 1) is true.] Therefore, P (k + 1) is true.Thus, by the PSMI, P (n) is true for all integers n ≥ a.

proof by ordinary induction skeleton proof

Proof. We proceed by induction.(i) Base step: [Verify that P (a) is true. This often, but not always, amounts to plugging n = a into two sides of some claimed equation and verifying that both sides are actually equal.](ii) Inductive step: [Your goal is to prove "For all k ∈ Z, if P (k) is true, then P (k + 1) is true."] Let k ≥ a be an integer and assume that P (k) is true. [Do something to derive that P (k + 1) is true.] Therefore, P (k + 1) is true.Thus, by the PMI, P (n) is true for all integers n ≥ a.

free variable

Recall that sentence "x > 0" is not itself a statement because we haven't specified whatx is. In other words, x is a this

set builder notation

S = {x ∈ A | x satisfies some condition}

proof by contradiction

Suppose that we want to prove some statement P (which might be something likeA → B or could be something even more complicated), assume ¬P is true and to then logically deduce a contradiction of the form Q∧¬Q, where Q is some statement (that is possibly equal to P ), the assumption that ¬P was true must have been erroneous, so P must in fact, be true, Proof: For the sake of a contradiction, assume ¬P .... [Use definitions and known results to derive some Q and its negation ¬Q.] ...This is a contradiction. Therefore, P.

empty set

The set containing no elements, denoted by the symbol ∅

universal quantifier

The symbol ∀ denotes "for all

existential quantifier

The symbol ∃ denotes "there exists"

set-equality proof via double-inclusion

To prove that A = B, prove that A ⊆ B and that B ⊆ A. Use the method outlined above for the set-inclusion proofs

set-inclusion proof

To prove that A ⊆ B, let x ∈ A and show that x ∈ B

what is means for two sets to be equal

Two sets A and B are _____________, denoted A = B , iff A ⊆ B and B ⊆ A

truth table logical equivalence

Two statements P and Q, expressed symbolically as P ↔ Q and read "P if and only if Q", if and only if they have the same truth table, the notation ≡ was used for logical equivalence, as in P ≡ Q.

factor

When n divides m, we call n a ____________ of m

set

a collection of objects called elements

irrational

a real number that cannot be expressed as a ratio of two integers

statement

a sentence that is either true or false, but not both

perfect square

an integer n, if and only if there exists an integer k such that n = k^2

Russell's paradox

arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox

direct proof

assume original statement, show that it is true using definitions, etc.

proof by arguing the contrapositive

assume the contrapositive, prove that it is true, therefore the original statement is true

cartisian product

f X and Y are sets, the ______________________ of X and Y is defined by X × Y = {(x,y) | x ∈ X,y ∈ Y }. That is, X × Y is the set of all ordered pairs where the first element in the pair is from X and the second element is from Y . The set X × X is sometimes denoted by X^2 . We similarly define the ___________________ of n sets, say X1 ,...,Xn, by big n thing to n and i=1 of Xi = X1 × ··· × Xn = {(x1 ,..., xn) | each xi ∈ Xi }

how to use euclidean algorithm to find GCD of two integers

gcd(128, 36): 128 = 36(3) + 20 36 = 20(1) +16 20 = 16(1) + ->4<- 16 = 4(4) + 0 4 is the gcd

postiive

if and only if 0 < x

negative

if and only if x < 0

logical not

is true if and only if A is false; expressed symbolically as ¬Aand called the negation of A

logical or

is true if and only if at least one of A or B is true; expressed symbolically as A ∨B and called the disjunction of A and B

conditional statement

is true if and only if both A and B are true, or A is false; expressed symbolically as A → B and called a conditional statement. Note that A → B may also be read as "A implies B" or "A only if B", A → B is logically equivalent to the disjunction¬A ∨B

logical and

is true if and only if both A and B are true; expressed symbolically as A ∧ B and called the conjunction of A and B.

complement

of A (relative to U) is the set A c = U \ A = {x ∈ U | x < A}.

set difference

of the sets A and B is A \ B = {x ∈ U | x ∈ A and x < B}.

intersection

of the sets A and B is A ∩B = {x ∈ U | x ∈ A and x ∈ B}.

union

of the sets A and B is A ∪B = {x ∈ U | x ∈ A or x ∈ B}

counterexample

provide an example in which the statement is false

purpose of strong induction and how it is different from ordinary induction

provides us with more information to use when trying to prove the statement, since we are proving more information it is stronger, different in that, The difference between weak induction and strong induction only appears in induction hypothesis. In weak induction, we only assume that particular statement holds at k-th step, while in strong induction, we assume that the particular statement holds at all the steps from the base case to k-th step

quotient

q in division algorithm

remainder

r in division algorithm

logical equivalence with truth tabes

show truth tables are equal to prove logical equivalence

how to find equivalence classes and why they make partition

start by plugging in x equal 0, 1, 2,... look for patterns, find partition

paradox

statements that run counter to one's intuition, sometimes in simple, playful ways, and sometimes in extremely esoteric and profound ways

how to solve division algorithm

suppose m = 27 and n = 5 => 27 = 5(q) + r => 27 = 5(5) + 2

existence proofs

there exists an x that satisfies some P(x)

purpose of ordinary induction

used to show a mathematical statement is true for all integers on some given interval

how to find values for s and t

using euclidean algorithm, gcd(a, b0 = sa + tb

how to check if reflexive, symmetric, or transitive

write proof to show they are true

subtraction

x - y is defined as x + (-y)

it is true that

¬(1 = 0)

symmtetric

∼ is _____________ if for all x,y ∈ A, if x ∼ y, then y ∼ x

transitive

∼ is _________________ if for all x,y, z ∈ A, if x ∼ y and y ∼ z, then x ∼ z

reflexive

∼ is __________________ if for all x ∈ A, x ∼ x (every element is related to itself).