proof defintions

Statement

a declarative sentence that must have a definite truth value, either true or

false but not both

Condtional Statement

A statement that can be written in

the form “If P then Q,” where P and Q are sentences.

Even Integer

An integer a provided that there exists an

integer n such that a= 2n.

Odd Integer

A integer a provided that there exists an integer n such that a= 2n+1

Pythgorean triple

Three natural numbers a, b, and c with a < b < c

are said to form a provided that a²+b²=c²

Logically equivalent statements

They have the same truth value for all possible combinations of truth values for all variables appearing in the two expressions.

Converse of P → Q

Q →P

Contrapositive of P → Q

~Q → ~P

Universal set for a variable

It is the set of specified objects from which objects may be chosen to substitute for the variable

Constant

a specific member of the universal set.

Equal sets

They have precisely the same elements.

Open Sentence

A sentence P (x_1, x_2, …, x_n) involving variables x_1, x_2,…, x_n with the property that when specific values from the universal set are assigned to x_1, x_2, …, x_n, then the resulting sentence is either true or false. That is, the resulting sentence is a statement.

Predicate

Another word for open sentence

Truth set of a predicate

the collection of objects in the universal set that can be substituted for the variable to make the predicate a true statement.

Universal quantifier

The phrase “for every” (or its equivalents) also denoted by (∀)

Existential quantifier

The phrase “there exists” (or its equivalents) also denotes by (∃)

Empty set

When a set contains no elements

Counterexample

It is an example that proves that (∀x) [p(x)] is a false statement, and hence its negation (∃x) ~[p(x)] is a true statement.

Perfect square

A natural number n for which there exists a natural number k such that n= k².

Prime number

A natural number p that is greater than 1 and the only natural numbers that are factors of p are 1 and p.

Composite number

A natural number other than 1 (1 is neither) that is not a prime number

Divides

When there exists a nonzero integer m and an integer n provides that there is an integer q such that n = mq.

What else can we say about m if it divides n.

m is a divisor of n, m is a factor of n, and n is a multiple of m

Proof

a convincing argument that some mathematical statement is true

Axiom

A mathematical statement that is accepted without proof.

Definition

Simply an agreement as to the meaning of a particular term. Usually made because a certain property is observed to occur frequently

Conjecture

A statement that we believe is plausible. That is, we think it is true, but we have not yet developed a proof that it is true.

Theorem

A mathematical statement for which we have a proof.

Proposition

A term that is often considered to be synonymous with “theorem”

Lemma

A true mathematical statement that was proven mainly to help in the proof of some theorem.

Corollary

A theorem that is easily proven once some other theorem has been proven.

Congruence modulo n

Let n be a natural number. If a and b are integers, then we say that provided that n divides a-b. (a-b = nk for some integer k)

Tautology

A compound statement S that is true for all possible combinations of truth values of the component statements that are part of S.

Contradiction

A compound statement that is false for all possible combinations of truth values of the component statements that are part of S

Absolute value

= { x if x≥0, -x if x<0