Converse, Inverse, and Contrapositive

1. 3.4 – More on the Conditional Converse, Inverse, and Contrapositive • p → q • If p, then q • q → p • If q, then p • Converse • If not p, then not q • Inverse • ̴ p → ̴ q • If not q, then not p • Contrapositive • ̴ q → ̴ p

2. 3.4 – More on the Conditional Determining Related Conditional Statements Given the conditional statement, determine the following: a) the converse, b) the inverse, and c) the contrapositive. If I live in Wisconsin, then I shovel snow, a) Converse If I shovel snow, then I live in Wisconsin. b) Inverse If I do not live in Wisconsin, then I do not shovel snow. c) Contrapositive If I do not shovel snow, then I do not live in Wisconsin.

3. 3.4 – More on the Conditional Equivalences A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent. Alternative Forms of “If p, then q” The conditional p → q can be translated in any of the following ways: If p, then q. p is sufficient for q. If p, q. q is necessary for p. p implies q. All p are q. ponly if q. q if p.

4. 3.4 – More on the Conditional Rewording Conditional Statements Write each statement in the form “if p, then q.” a) You’ll be sorry if I go. (q if p) If I go, then you’ll be sorry. b) Today is Sunday only if yesterday was Saturday. (p only if q) If today is Sunday, then yesterday was Saturday. c) All Chemists wear lab coats. (All p are q) If you are a Chemist, then you wear a lab coat.

5. 3.4 – More on the Conditional Negation of a Conditional The negation of A Conditional as a Disjunction The conditional is equivalent to Examples: If the river is narrow, then we can cross it. p: the river is narrow. p: the river is not narrow. q: we can cross it. q: we cannot cross it. Negation: Disjunction: The river is narrow and we cannot cross it. The river is not narrow or we can cross it.

6. 3.4 – More on the Conditional Negation of a Conditional The negation of A Conditional as a Disjunction The conditional is equivalent to Examples: If your are absent, then you have a test. p: you are absent. p: you are not absent. q: you do not have a test. q: you have a test. Negation: Disjunction: You are absent and you do not have a test. You are not absent or you have a test.