Math 381

Binary Relation

Let A and B be sets. A .... from A to B is a subset of A x B.

Modus Ponens

p p->q .: q

Conjunction

p q .: p and q

Hypothetical syllogism

p -> q q -> r .: p -> r

Disjunctive syllogism

p V q not q .: q

Resolution

p V q not p V r .: q V r

Simplification

p and q .: p

Proof by Contradiction

try to find a contradiction q such that not p -> q. Because q is false but not p -> q is true we can conclude that not p is false, which means p is true

Proof by Contraposition

use p -> q = not q -> not p which is it's contrapositive.

Relatively Prime

The integers a and b are .... if their greatest common divisor is 1.

Prime

An integer p greater than 1 is called .... if the only positive factors of p are 1 and p.

Inductive Hypothesis

(IH) The assumption that P(k) is true.

Combinatorial Proof

A ... ... of an identity is a proof that uses counting arguments to prove that both sides of the identity count the same objects but in different ways or a proof that is based on showing that there is a bijection between the sets of objects counted by the two sides of the identity. These two types of proofs are called double counting proofs and bijective proofs, respectively.

Set

A .... is an unordered collection of objects called elements or members of the ... A ... is said to contain its elements.

Direct Proof

A .... of a conditional statement p→q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A ..... shows that a conditional statement p→q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. In

Relation

A .... on a set A is a relation from A to A.

Symmetric Relation

A Relation on a set A is called .... if (b,a) is an element of the relation R whenever (a,b) is an element of R, for all

onto definition

A function f from A to B is called ...., or a surjection, if and only if for every element b∈B there is an element a∈A with f (a) = b. A function f is called surjective if it is .....

one to one

A function f is said to be ...., or an injunction, if and only if f (a) = f (b)implies that a = b for all a and b in the domain of f. A function is said to be injective if it is .....

composite

A positive integer that is greater than 1 and is not prime is called .....

Reflexive

A relation R on a set A is called .... if (a,a) it is an element of the relation for every element a that is an element of set A.

Transitive Relation

A relation R on a set A is called .... if whenever (a,b) are elements of the relation R and (b,c) are also elements of this relation, then (a,c) is an element of R for all a,b,c that are elements of A.

Equivalence Relation

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

Antisymmetric Relation

A relation r on a set A such that for all a,b that are elements of A, if (a,b) is an element of the relation r and (b,a) is an element of the relation R then a=b is called ....

argument

An .... in propositional logic is a sequence of propositions. All but the final proposition in the ...... are called premises and the final proposition is called the conclusion. An .... is valid if the truth of all its premises implies that the conclusion is true. An .... form in propositional logic is a sequence of compound propositions involving propositional variables. An .... form is valid no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true.

The fundamental theorem of arithmetic

Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are written in order of nondecreasing size.

Existential instantiation

Existential Quantifier P(x) .: P(c) for some element c

Uniqueness Quantifier

Existential Quantifier followed by factorial symbol. Means "there exists a unique x such that P(x) is true."

Power Set Definition

Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S).

Congruence/Modulus

If a and b are integers and m is a positive integer, then a is congruent to b modulo m if m divides a - b. We use the notation a = b(mod m) to indicate that a is congruent to b modulo m. We say that a = b(mod m) is a congruence and that m is it's modulus (plural moduli).

a | b

If a and b are integers with a not equal to 0, we say that ..... if there is an integer c such that b = ac, or equivalently, if b/a is an integer. When .... we say that a is a factor or divisor of b, and that b is a multiple of a. The notation a | b denotes that .....

Image

Let f be a function from A to B and let S be a subset of A. The .... of S under the function f is the subset of B that consists of the images of the elements of S.

Bezout's Coefficients/Identity

If a and b are positive integers, then integers s and t such that gcd(a, b) = sa + tb are called Bézout ..... of a and b (after Étienne Bézout, a French mathematician of the eighteenth century). Also, the equation gcd(a, b) = sa + tb is called Bézout's ....

Bezout's Theorem

If a and b are positive integers, then there exist integers s and t such that gcd(a, b) = sa + tb.

The Sum Rule

If a task can be done either in one of n1 ways or in one of n2 ways, where none of the set of n1 ways is the same as any of the set of n2 ways, then there are n1 + n2 ways to do the task.

Bezout's Lemma

If a, b, and c are positive integers such that gcd(a, b) = 1 and a | bc, then a | c.

Divides by Theorem Corollary

If a,b, and c are integers where a does not equal 0, such that a | b and a | c, then a | mb + nc whenver m and n are integers

domain/codomain/image/preimage

If f is a function from A to B, we say that A is the .... of f and B is the ..... of f. If f (a) = b, we say that b is the ..... of a and a is a ..... of b. The range, or image, of f is the set of all images of elements of A. Also, if f is a function from A to B, we say that f maps A to B.

r-permutations theorem corollary 1

If n and r are integers with 0 <= r <= n, then P(n,r) = n!/(n-r)!

Prime Divisor Theorem

If n is a composite integer then n has a prime divisor less than or equal to the square root of n.

r-permutations theorem

If n is a positive integer and r is an integer with a <= r <= n, then there are P(n,r) = n(n-1)(n-2) .... (n-r+1) r-permutations of a set with n distinct elements.

Bezout's Lemma 2

If p is a prime and p | a1a2 · · · a n, where each ai is an integer, then p | ai for some i.

Definitions of Division Algorithm

In the equality given in the division algorithm, d is called the divisor, a is called the dividend, q is called the quotient and r is called the remainder. The notatioin is used to express the quotient and remainder q = a div d, r = a mod d.

Method of Successive Substitutions

In the final presentation of such a solution, one starts out with one side of the proposed logical equivalence, and changes one part of it at a time.

Function Definition

Let A and B be nonempty sets. A .... f from A to B is an assignment of exactly one element of B to each element of A. We write ... = b if b is the unique element of B assigned by the ....f to the element a of A.

Difference of Sets Definition

Let A and B be sets. The .... of A and B, denoted by A- B, is the set containing the elements that are in A but not in B. The .... is also called the complement of B with respect to A.

Equivalence Class

Let R be an equivalence relation on a set A. The set of all elements that are related to an element of A is called an .... .... of a. The .... .... of a with respect to R is dentated by [a]R. When only one relation is under consideration we can delete the subscript R and write [a] for this ... .... .

Theorem for equivalence

Let R be an equivalence relation on a set A. These statements for elements a and b are equivalent. (i) aRb (ii) [a] = [b] (iii) [a] union [b] does not equal the empty set.

Theorem for Partitions

Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, given a partition { Ai | i is an element of I} of the set S, there is an equivalence relation R that has the sets Ai, i is an element of I, as its equivalence classes.

Euclidean Algorithm Lemma

Let a = bq + r, where a, b, q, and r are integers. Then gcd(a, b) = gcd(b, r).

a = b (mod m) theorem

Let a and b be integers and let m be a positive integer. Then .... if and only if a mod m = b mod m.

Greatest Common Divisor

Let a and b be integers, not both zero. The largest integer d such that d | a and d | b is called the ..... of a and b. The greatest common divisor of a and b is denoted by gcd(a, b).

GCD/LCM Theorem

Let a and b be positive integers. Then ab = gcd(a, b) · lcm(a, b).

Division Algorithm/Theorem

Let a be an integer and d a positive integer. There are unique integers q and r with 0 <= r < d, such that a = dq + r.

Divides by Theorem part 1

Let a,b, and c be integers, where a does not equal 0. Then if a | b and a | c, then a | (b+c)

Divides by Theorem part 3

Let a,b, and c be integers, where a does not equal 0. Then if a | b and b | c, then a | c.

Divides by Theorem part 2

Let a,b, and c be integers, where a does not equal 0. Then if a | b, then a | bc for all integers c

Inverse Function

Let f be a one-to-one correspondence from the set A to the set B. The .... function of f is the function that assigns to an element b belonging to B the unique element a in A such that f (a) = b. The .... function of f is denoted by f−1 . Hence, f−1 (b) = a when f (a) = b.

Function Properties

Let f1 and f2 be functions from A to R. (f1 + f2)(x) = f1(x) + f2(x), (f1 f2)(x) = f1(x)f2(x).

Composition function definition

Let g be a function from the set A to the set B and let f be a function from the set B to the set C. The .... of the functions f and g, denoted for all a∈A by f◦g, is defined by (f◦g)(a) = f (g(a)).

mod m corollary

Let m be a positive integer and let a and b be integers then (a + b) mod m = ((a mod m) + (b mod m)) mod m and ab mod m = ((a mod m)( b mod m) mod m.

GCD/Mod Theorem

Let m be a positive integer and let a, b, and c be integers. If ac≡bc (mod m) and gcd(c, m) = 1, then a≡b (mod m).

mod m theorem 2

Let m be a positive integer. If a is congruent b modulo m and c is congruent d modulo m, then a + c is congruent b+d modulo m and a*c is congruent modulo b*d modulo m.

Congruent modulo m theorem

Let m be a positive integer. The integers a and b are ... if and only if there is an integer k such that a = b + km.

Theorem 2/Pascal's Identity

Let n and k be positive integers with n >= k. Then c(n+1,k) = c(n, k-1) + c(n,k).

Combinations corollary

Let n and r be non-negative integers with r <= n. Then C(n,r) = C(n, n-r).

Binomial Theorem Corollary 1

Let n be a non-negative integer. Then sum from k= 0 to n c(n,k) = 2^n.

Binomial Theorem Corollary 3

Let n be a non-negative integer. Then the sum of k=0 to n of 2^k c(n,k) = 3^n .

Binomial Theorem Corollary 2

Let n be a positive integer. Then sum of k=0 to n of (-1)^k c(n,k) = 0.

Binomial Theorem

Let x and y be variables, and let n be a non-negative integer. Then (x+y)^n = sum c(n,j) x^(n-j) y^j = c(n,0)x^n + c(n,1) x^(n-1)y + ... + c(n,n) y^n.

Universal generalization

P(c) for an arbitrary c .: Universal quantifier P(x)

Existential Generalization

P(c) for some element c .: Existential Quantifier P(x)

logically equivalent

Statements involving predicates and quantifiers are ... .... if and only if they have the same truth value no matter which predicates are substituted into these statements and which domain of discourse is used for the variables in these propositional functions.

The Product Rule

Suppose that a procedure can be broken down into a sequence of two tasks. If there are n1 ways to do the first task and for each of these ways of doing the first task, there are n2 ways to do the second task, then there are n1*n2 ways to do the procedure.

Existential Quantification

The .... .... of P(x) is the proposition. "There exists an element x in the domain such that P(x)."

Universal Quantification

The .... ..... of P(x) is the statement. "P(x) for all values of x in the domain." The Notation .... denotes the ..... of P(x). We read ... as "for all xP(x)" or "for every xP(x)."

Intersection of Collection Definition

The .... of a collection of sets is the set that contains those elements that are members of all the sets in the collection.

Union of Collection Definition

The .... of a collection of sets is the set that contains those elements that are members of at least one set in the collection.

Least common Multiple

The ..... of the positive integers a and b is the smallest positive integer that is divisible by both a and b. The .... of a and b is denoted by ...(a, b).

r-combinations with repetition theorem

The are C(n+r -1, r) = C(n+r-1, n-1) r-combinations from a set with n elements when repetition of elements is allowed.

Bijection

The function f is a one-to-one correspondence, or a ...., if it is both one-to-one and onto. We also say that such a function is .....

Even or Odd

The integer n is .... if there exists an integer k such that n = 2k. n is ... if there exists an integer k such that n = 2k + 1.

r-combinations theorem

The number of r-combinations of a set with n elements, where n is a non-negative integer and r is an integer with 0 <= r <= n, equals C(n,r) = n!/r!(n-r)!

r-permutations with repetition theorem

The number of r-permutations of a set of n objects with repetition allowed is n^r.

Rational or Irrational

The real number r is .... if there exist integers p and q with q = 0 such that r = p/q. A real number that is not rational is called .....

Subset Definition

The set A is a ... of B if and only if every element of A is also an element of B.

Showing Two sets are equal

To show that two sets A and B are equal, show that A is a subset of B and that B is a subset of A.

Equivalent

Two elements a and b that are related by an equivalence relation are called .... . The notation a ~ b is often used to denote that a and b are equivalent elements with respect to a particular equivalence relation.

Disjoint Defintion

Two sets are called disjoint if their intersection is empty

Set Equality

Two sets are said to be if and only if they have the same elements.

Universal instantiation

Universal quantifier P(x) .: P(c)

Set Builder notation

We will characterize all those elements in a set by stating the property or properties they must have to be members.

De Morgan's Laws

not ( p and q) equivalent to not p or not q not (p or q) equivalent to not p and not q

Modus tollens

not q p -> q .: not p

Addition

p .: p V q