3.3 Truth Tables for the Conditional and Biconditional 121 Suppose I said to you, “If you get an A, then I will buy you a car.” As we discussed in Section 3.1, this statement is called a conditional statement. In this section, we will discuss under what conditions a conditional statement is true and under what conditions a conditional statement is false. SECTION 3.3 Truth Tables for the Conditional and Biconditional LEARNING GOALS Upon completion of this section, you will be able to: 7 Construct truth tables involving conditional statements. 7 Construct truth tables involving biconditional statements. 7 Understand self-contradictions, tautologies, and implications. Why This Is Important We often encounter statements of the form if ... then and if and only if. These statements are called conditional statements and biconditional statements, respectively. Such statements can be found in advertising claims, political speeches, and many legal documents. Understanding when these statements are true and when these statements are false can have a large impact on our lives. For example, when purchasing a home, it is very important that you understand the statements contained in the real estate contract. Conditional In Section 3.1, we mentioned that the statement preceding the conditional symbol is called the antecedent and that the statement following the conditional symbol is called the consequent. For example, consider ∨ → ∼ ∧ p q q r ( ) [ ( )]. In this statement, p q ( ) ∨ is the antecedent and q r [ ( )] ∼ ∧ is the consequent. To develop a truth table for the conditional statement, consider the statement “If you get an A, then I will buy you a car.” Assume this statement is true except when I have actually broken my promise to you. Let p q : You get an A. : I buy you a car. 81. Must p q r ( ) ∧ ∼ ∨ and q r p ( ) ∧ ∼ ∨ have the same number of trues in their answer columns? Explain. Yes Research Activity 82. Logic and Set Theory Do research and write a report on each of the following. a) The relationship between negation in logic and complement in set theory b) The relationship between conjunction in logic and intersection in set theory c) The relationship between disjunction in logic and union in set theory 80. On page 114, we indicated that a compound statement consisting of n simple statements had 2n distinct true–false cases. a) How many distinct true–false cases does a truth table containing simple statements p q r , , , and s have? 16 b) List all possible true–false cases for a truth table containing the simple statements p q r , , , and s. * c) Use the list in part (b) to construct a truth table for q p r s ( ) ( ). ∧ ∨ ∼ ∧ * d) Construct a truth table for r s p q ( ) ( ). ∼ ∧ ∼ ∧ ∼ ∨ * *See Instructor Answer Appendix Lopolo/123RF
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