notes/docs/mathematics/logic.md

Logic

Definition: a statement is a sentence that is either true or false, never both.

Definition - Logical operators: let A and B be assertions.

• The assertion A and B (A \land B) is true, iff both A and B are true.
• The assertion A or B (A \lor B) is true, iff at least one of A and B is true.
• The negation of A (\neg A) is true iff A is false.

Definition - Implies: if A and B are assertions then the assertion if A then B (A \implies B) is true iff

• A is true and B is true,
• A is false and B is true,
• A is false and B is false.

This also works the opposite way, if B then A (A \Longleftarrow B)

Definition - If and only if: if A and B are assertions then the assertion A if and only if B (A \iff B) is true iff

• (A \Longleftarrow B) \land (a \implies B).

This leads to the following table.

A	B	A \implies B	A \Longleftarrow B	A \iff B
true	true	true	true	true
true	false	false	true	false
false	true	true	false	false
false	false	true	true	true

Definition: suppose P and Q are assertions. P implies Q if P \implies Q is true. P and Q are equivalent if P implies Q and Q implies P.

Methods of proof

Direct proof: for proving P \implies Q only consider the case where P is true.

Proof by contraposition: proving P \implies Q to be true by showing that \neg Q \implies \neg P is true.

Proof by contradiction: using the equivalence of P \implies Q and \neg Q \implies \neg P by assuming P is not true and deducing a contradiction with some obviously true statement Q.

Proof by cases: dividing a proof into cases which makes use of the equivalence of (P \lor Q) \implies R and (P \implies R) \land (Q \implies R). Which together cover all situations under consideration.