Exam Three: Proofs

Theorem

A mathematical assertion that can be shown to be true

Axiom

A statement that is assumed to be true and that can be used as a basis for proving theorems

Proof

A demonstration that a theorem is true

Lemma

A theorem used to prove other theorems

Corollary

A proposition that can be proved as a consequence of a theorem that has just been proved

Conjecture

A mathematical assertion proposed to be true, but that has not been proved

Direct Proof

A proof that p → q is true that proceeds by showing that q must be true when p is true

Indirect Proof/Proof by Contraposition

A proof that p → q is true that proceeds by showing that p must be false when q is false

Proof by Contradiction

A proof that p is true based on the truth of the conditional statement ¬p → q, where q is a contradiction

Vacuous Proof

A proof that p → q is true based on the fact that p is false

Proof by Cases

A proof broken into separate cases, where these cases cover all possibilities

Existence Proof

A proof of the form ∃xP(x)

Exhaustive Proof

A proof that establishes a result by checking a list of all possible cases