MAT 243 Proofs

Proofs

sequence of valid arguments, rules of inference, that establishes the truth
of a mathematics statement

Axioms

self-evident statements

Theorem

an important mathematical true statement that have been proved

Proposition

less important statement 

Lemma

a small theorem that are used to prove important more complicated
theorem, are building blocks, or puzzle pieces that are used in the proof
of a theorem

Corollary 

consequence or a special case of a theorem

Direct proofs

Prove the statement p → q.
Assume p and prove the conclusion q.

Proof by contrapositive

Prove the statement p → q.
Prove ¬q → ¬p. Use direct proof techniques.

Proof by contradiction

Prove the statement p. Assume the statement p is false,
that is assume ¬p is true. Derive a contradiction. Therefore ¬! is false. Thus, ! is
true. Note the p is not necessarily a conditional statement here.

Definition of even integers 

An integer n is said to be even if and only if there is
an integer k such that n = 2k.

Definition of odd integers 

An integer n is said to be odd if and only if there is an
integer k such that n = 2k + 1.

Definition of rational numbers

A real number r is said to be rational if there exist
integers p and q, q ≠ 0, such that r = p/q.

Definition of irrational numbers 

not rational