Untitled Flashcards Set

Argument

A sequence of statements leading to a conclusion.

Valid Argument

A conclusion that logically follows from true hypotheses.

Proof

A valid argument.

Fallacy

An invalid argument.

Axiom (Postulate, Law)

A statement assumed to be true.

Conjecture

A statement proposed to be true.

Theorem

A statement proven to be true.

Lemma

A preliminary theorem.

Corollary

A theorem that follows directly from another theorem.

Modus Ponens

If p→q and p is true, then q is true.

Universal Modus Ponens

If 'All x have property P', and a is an x, then a has property P.

Universal Modus Tollens

If 'All x have property P', but a does not have P, then a is not an x.

Vacuous Proof

Proving p→q by showing p is false.

Trivial Proof

Proving p→q by showing q is always true.

Direct Proof

Assuming hyposthesis and logically deriving conclusion.

Indirect Proof (Contraposition)

Assume the conclusion is false, and show this means

the hypothesis is false (i.e. Prove the contrapositive.)
(Proving p→q by proving ¬q→¬p.)

Proof by Contradiction

To show P is true, assume P is false, and show this

results to a contradiction (i.e. a false statement).is not possible, thereby proving P must be true.

Proof of Equivalence

Proving p↔q by proving both p→q and q→p.

Proof by Counterexample

Showing that a universal claim is false by finding a counterexample.
(“For all x, P(x)” is false, find a c where P(c)=false)

Constructive Proof

Explicitly finding an example to prove existence.

Non-Constructive Proof

Proving existence without identifying an example.
(Show that for every positive integer n,

there is a prime bigger than n.)

Uniqueness Proof

Proving that a solution exists and is unique.

Exhaustive Proof

Checking all possible cases.

Proof by Cases

Breaking a proof into multiple scenarios.

Pigeonhole Principle

If k+1 objects are placed into k containers, at least one container has more than one object.