104 CHAPTER 3 Logic If and Only If Statements The biconditional is symbolized by ↔ and is read “if and only if.” The phrase if and only if is sometimes abbreviated as “iff.” The statement ↔p q is read “p if and only if q.” Example 8 Write Statements Using the Biconditional Let p q : Alex plays goalie on the lacrosse team. : The Huskies win the Champion’s Cup. Write the following symbolic statements in words. Marcelo Murillo/ Shutterstock Now try Exercise 65 You will learn later that ↔p q means the same as → ∧ → p q q p ( ) ( ). Therefore, the statement “I will go to college if and only if I can pay the tuition” has the same logical meaning as “If I go to college then I can pay the tuition, and if I can pay the tuition then I will go to college.” A summary of the connectives discussed in this section is given in Table 3.1. Table 3.1 Logical Connectives Formal Name Symbol Read Symbolic Form Negation ∼ “Not” ∼p Conjunction ∧ “And” ∧ p q Disjunction ∨ “Or” ∨ p q Conditional → “If–then” →p q Biconditional ↔ “If and only if” ↔p q Learning Catalytics Keyword: Angel-SOM-3.1 (See Preface for additional details.) Instructor Resources for Section 3.1 in MyLab Math • Objective-Level Videos 3.1 • Interactive Concept Video: Applying Four Rules of Logic • PowerPoint Lecture Slides 3.1 • MyLab Exercises and Assignments 3.1 Exercises Warm Up Exercises In Exercises 1–6, fill in the blanks with an appropriate word, phrase, or symbol(s). 1. A sentence that can be judged either true or false is called a(n) ____________. Statement 2. A statement that conveys only one idea is called a(n) ____________ statement. Simple 3. A statement that consists of two or more simple statements is called a(n) ____________ statement. Compound 4. Words such as all, none (or no), and some are examples of ____________. Quantifiers 5. a) The negation is symbolized by ∼ and is read “____________.” Not b) The conjunction is symbolized by ∧ and is read “____________.” And c) The disjunction is symbolized by ∨ and is read “____________.” Or SECTION 3.1 a) ↔p q b) q p ↔∼ c) p q ( ) ∼ ↔∼ Solution a) Alex plays goalie on the lacrosse team if and only if the Huskies win the Champion’s Cup. b) The Huskies win the Champion’s Cup if and only if Alex does not play goalie on the lacrosse team. c) It is false that Alex plays goalie on the lacrosse team if and only if the Huskies do not win the Champion’s Cup. 7
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