Validating Statements

Recognising a statement is only half the task; the other half is deciding whether it is true. Doing this carefully depends on identifying exactly which special words appear in the statement — "and"/"or", "if-then"/"if and only if", "for every"/"there exists" — because each comes with its own rule for validation.

Rule 1 — "p and q". To show "p and q" is true, prove p is true, and separately prove q is true. Both steps are required.

Rule 2 — "p or q". To show "p or q" is true, it is enough to handle either of these cases: assume p is false and show q must then be true; or assume q is false and show p must then be true.

Rule 3 — "if p then q". Either of two approaches proves this: the direct method — assume p is true and show q must follow; or the contrapositive method — assume q is false and show p must then be false. (The contrapositive method is developed further in the next subsection.)

Rule 4 — "p if and only if q". Both directions must be shown separately: (i) if p is true, then q is true, and (ii) if q is true, then p is true. …