VARIATIONS OF THE CONDITIONAL AND IMPLICATIONS 3.4 2
VARIATIONS OF THE CONDITIONAL 3 Variations of p q Converse: Inverse: Contrapositive: p q is logically equivalent to ~ q ~ p q p is logically equivalent to ~ p ~ q
EXAMPLES OF VARIATIONS 4 Converse: q p p: n is not an even number. q: n is not divisible by 2. Conditional: p q If n is not an even number, then n is not divisible by 2. Contrapositive: ~q ~q ~p If n is divisible by 2, then n is an even number. Inverse: ~ p ~ q If n is an even number, then n is divisible by 2. If n is not divisible by 2, then n is not an even number.
ConditionalConverseInverseContrapositive pq TT TF FT FF p q T F T T q p T T F T ~q ~p T F T T ~p ~ q T T F T Equivalent Equivalent
CONDITIONAL EQUIVALENTS 6 StatementEquivalent forms if p, then q p q p is sufficient for q q is necessary for p p only if q q if p
BICONDITIONAL EQUIVALENTS StatementEquivalent forms p if and only if q p q p is necessary and sufficient for q q is necessary and sufficient for p q if and only if p 7
EXAMPLES OF VARIATIONS 8 Given: h: honk u: you love Ultimate Write the following in symbolic form. Honk if you love Ultimate. If you love Ultimate, honk. Honk only if you love Ultimate. A necessary condition for loving Ultimate is to honk. A sufficient condition for loving Ultimate is to honk To love Ultimate, it is sufficient and necessary that you honk. or
TAUTOLOGIES AND CONTRADICTIONS A statement that is always true is called a tautology. A statement that is always false is called a contradiction. 9
EXAMPLE 10 p~p p ~ p TFF FTF Show by means of a truth table that the statement p ~ p is a contradiction.
IMPLICATIONS 11 The statement p is said to imply the statement q, p q, if and only if the conditional p q is a tautology.
EXAMPLE 12 pqp q(p q) Λ pq[(p q) Λ p] q TTTTTT TFFFFT FTTFTT FFTFFT END Show that