mathematics-physics-wiki/docs/en/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.