2 Variations of the Conditional and Implications 3.4
3 Variations of the conditional Variations of p → qConverse:Inverse:Contrapositive:p → q is logically equivalent to ~ q → ~ pq → p is logically equivalent to ~ p → ~ q
4 Examples of variations p: n is not an even number.q: n is not divisible by 2.Conditional: p → qIf n is not an even number, then n is not divisible by 2.Converse: q → pIf n is not divisible by 2, then n is not an even number.Inverse: ~ p → ~ qIf n is an even number, then n is divisible by 2.Contrapositive: ~q → ~pIf n is divisible by 2, then n is an even number.
5 p q T F p → q T F q → p T F ~p → ~ q T F ~q → ~p T F Equivalent ConditionalConverseInverseContrapositivepqTFp → qTFq → pTF~p → ~ qTF~q → ~pTF
6 Conditional Equivalents StatementEquivalent formsif p, then q p → qp is sufficient for qq is necessary for pp only if qq if p
7 Biconditional Equivalents StatementEquivalent formsp if and only if qp ↔ qp is necessary and sufficient for qq is necessary and sufficient for pq if and only if p
8 Examples of variations Given: h: honku: you love UltimateWrite 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 honkTo love Ultimate, it is sufficient and necessary that you honk.or
9 Tautologies and Contradictions A statement that is always true is called a tautology. A statement that is always false is called a contradiction.
10 ExampleShow by means of a truth table that the statement p ↔ ~ p is a contradiction.p~pp ↔ ~ pTF
11 ImplicationsThe statement p is said to imply the statement q, p q, if and only if the conditional p → q is a tautology.