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3.4 More on the Conditional

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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 qIf p, then q. Converseq pIf q, then p. Inverse~p ~qIf not p, then not q. Contrapositive~q ~pIf not q, then not p.

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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 dont live in Miami, then I dont live in Florida. If I dont live in Florida, then I dont live in Miami.

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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. ConditionalConverseInverseContrapositive pqp qq p~p ~q~q ~p

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

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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.

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Common Translations of p q 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): – 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.

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

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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 pq p q TTT TFF FTF FFT

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Determining Whether Biconditionals Are True or False Determine whether each biconditional statement is true or false. – = 15 if and only if = 16 6 = 5 if and only if = 10 if and only if = 36 pq p q TTT TFF FTF FFT

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