3.4 Equivalent Statements 137 We summarize the result of Example 8 as follows. m Pickleballl Example 8 The Negation of a Conditional Statement Determine a statement equivalent to p q ( ). ∼ → Solution Begin with p q p q, → ⇔∼ ∨ negate both statements, and use De Morgan’s laws. p q p q p q p q p q ( ) ( ) → ⇔∼ ∨ ∼ → ⇔∼ ∼ ∨ ⇔ ∧ ∼ Negate both statements. De Morgan’s laws Therefore, p q ( ) ∼ → is equivalent to p q. ∧ ∼ 7 Now try Exercise 37 The Negation of the Conditional Statement Written as a Conjunction p q p q ( ) ∼ → ⇔ ∧ ∼ Example 9 Write an Equivalent Statement Write a statement that is equivalent to “It is not true that if I play pickleball then I win the tournament.” Solution Let p q : I play pickleball. : I win the tournament. The given statement can be represented symbolically as p q ( ). ∼ → We showed in Example 8 that p q ( ) ∼ → is logically equivalent to p q. ∧ ∼ Therefore, an equivalent statement is “I play pickleball and I do not win the tournament.” 7 Now try Exercise 41 Using the fact that p q p q ( ) , ∼ → ⇔ ∧ ∼ can you determine what p q ( ) ∼ →∼ is equivalent to as a conjunction? If you answered p q, ∧ you answered correctly. Variations of the Conditional Statement We know that p q → is equivalent to p q. ∼ ∨ Are any other statements equivalent to p q? → Yes, there are many. Now let’s look at the variations of the conditional statement to determine whether any are equivalent to the conditional statement. The variations of the conditional statement are made by switching and/or negating the antecedent and the consequent of a conditional statement. The variations of the conditional statement are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional. Learning Catalytics Keyword: Angel-SOM-3.4 (See Preface for additional details.) Listed here are the variations of the conditional with their symbolic form and the words we say to read each one. Variations of the Conditional Statement Variations of the Conditional Statement Name Symbolic form Read Conditional p q → “If p, then q” Converse of the conditional q p → “If q, then p” Inverse of the conditional p q ∼ →∼ “If not p, then not q” Contrapositive of the conditional q p ∼ →∼ “If not q, then not p” Lawrence Malvin/123RF
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