3.4 More on the Conditional

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3.4 More on the Conditional

Converse, Inverse, and Contrapositive If you interchange the antecedent and the consequent of a conditional, you form a new conditional known as the converse of the original statement. If you negate both the antecedent and the consequent, you form the inverse of the statement. If you interchange and negate the antecedent and consequent, you form the contrapositive of the statement. Related Conditional Statements Conditional (original) p  q If p, then q. Converse q  p If q, then p. Inverse ~p  ~q If not p, then not q. Contrapositive ~q  ~p If not q, then not p.

Determining Related Conditional Statements Given the conditional statement If I live in Miami, then I live in Florida determine each of the following: converse: inverse: contrapositive: If I live in Florida, then I live in Miami. If I don’t live in Miami, then I don’t live in Florida. If I don’t live in Florida, then I don’t live in Miami.

Truth Table of Related Conditional Statements Complete the truth table below to see how the truth values of related conditional statements compare. Notice that the conditional statement and its contrapositive are equivalent, while the converse and inverse are equivalent. Conditional Converse Inverse Contrapositive p q p  q q  p ~p  ~q ~q  ~p

Determining Related Conditional Statements For the conditional statement ~p  q, write each of the following: Converse Inverse Contrapostive

Alternative Forms of “If p, then q” Not every conditional statement is in if…then form. For example, the statement “If you are 18, then you can vote” could be written in the following ways: You can vote if you are 18. You are 18 only if you can vote. Being able to vote is necessary for you to be 18. All 18-year-olds can vote. Being 18 implies that you can vote.

Common Translations of p  q If p, then q. p is sufficient for q. If p, q. q is necessary for p. p implies q. All p are q. p only if q. q if p. The conditional p  q can be translated in any of the following ways (does NOT depend on truth/falsity of the statement or its components):

Rewording Conditional Statements Reword each statement in the form “if p, then q”. You’ll be sorry if I go. Today is Friday only if yesterday was Thursday. All nurses wear white shoes.

Biconditionals The compound statement “p if and only if q” (often abbreviated “p iff q”) is called a biconditional. Biconditionals are symbolized p  q, and are interpreted as the conjunction of the two conditionals p  q and q  p. So, by definition, p  q  (q  p)  (p  q), which leads to the following truth table: p if and only if q p q p  q T F

Determining Whether Biconditionals Are True or False Determine whether each biconditional statement is true or false. 6 + 9 = 15 if and only if 12 + 4 = 16 6 = 5 if and only if 12 ≠ 12 5 + 2 = 10 if and only if 17 + 19 = 36 p q p  q T F