Proofs/Definitions

Question 1

Conditional Statement: p → q
Assume that p is true.
Apply rules of inference, axioms, and logical equivalences to show/prove that q MUST
also be true.

Answer 1

Direct Proof

Question 2

Conditional Statement: p → q
Assume that ¬q is true. Apply rules of inference, axioms, and logical equivalences to show/prove that ¬p MUST also be true.

Answer 2

Contraposition

Indirect Proof

Question 3

Conditional Statement: p → q
Assume that
p is true and q is false
or
q is false and p is true
Apply rules of inference, axioms, and logical equivalences to contradict assumption such
that p is true and q is also true.

Answer 3

Contradiction

Indirect Proof

Question 4

Conditional Statement: P(x) → Q(x)
Find a real value of x that satisfies
P(x) but contradicts Q(x)

Answer 4

Counterexample

Proof Method

Question 5

Conditional Statement: P(x) → Q(x)
Breakup premise, P(x) → Q(x), into unique cases and prove each case, P(x) → Q(x), using any other
preferable Proof Method.

Answer 5

Rule of Cases

Proof Method

Question 6

A sequence of propositions

Answer 6

Argument

Question 7

An argument’s simplest form

Answer 7

Rule of inference

Question 8

The premises imply the conclusion

Answer 8

Valid argument

Question 9

An explanation via valid arguments that establishes the truth of a statement.

Answer 9

Proof

Question 10

A mathematical explanation with
regard to the meaning of a word/statement.

Answer 10

Definition

Question 11

a statement originally considered to be true.

Answer 11

Axiom

Question 12

A statement that can be proven to be true by using: definitions, theorems, axioms, or rules of inference.

Answer 12

Theorem

Question 13

A true statement or ‘auxiliary theorem’ used in proving other true statements.

Answer 13

Lemma

Question 14

A true statement or ‘auxiliary theorem’ that is a simple deduction from a theorem.

Answer 14

Corollary

Question 15

A statement that is being proposed to be true, but for which we have no proof.

Answer 15

Conjecture