3.3 Truth Tables for the Conditional and Biconditional 123 statement given in Table 3.16. Column 4 represents the truth values of the antecedent, and column 3 represents the truth values of the consequent. Remember that the conditional is false only when the antecedent is true and the consequent is false, as in cases (rows) 1, 2, and 4 of column 5. Now try Exercise 17 A truth table cannot by itself determine whether a compound statement is true or false. However, a truth table does allow us to examine all possible cases for compound statements. Biconditional The biconditional statement p q ↔ means that p q → and q p, → or, symbolically, p q q p ( ) ( ). → ∧ → To determine the truth table for p q, ↔ we will construct the truth table for p q q p ( ) ( ). → ∧ → In Table 3.19, we use column 3 and column 6 to obtain the answer in column 7. Table 3.20 shows the truth values for the biconditional statement. 7 MATHEMATICS TODAY Boolean Algebra Georgejmclittle/ Shutterstock Logic is part of the branch of mathematics known as Boolean algebra, named after George Boole (see Profile in Mathematics on page 101). Boolean algebra can be used to evaluate electrical circuits to determine whether current will flow through the circuit. This topic is developed further in Section 3.7. An electrical switch that allows current to flow is given a value of 1, and current flowing is considered a true statement. A switch that does not allow current to flow is given a value of 0, and this lack of current flowing is considered a false statement. Boolean algebra is used by engineers when designing virtually every modern electronic device including computers, tablets, smartphones, and other smart devices. These devices contain computer chips that are programmed using the not, and, or, if ... then , and if and only if statements we are studying in this chapter. In addition to Boolean algebra, computers and other electronic devices make use of the binary number system, which we discuss in Chapter 4. Why This Is Important Every day our modern society relies more and more on computers and electronic devices. Each of these devices contains computer chips that are programmed using Boolean algebra. Having a basic understanding of logic can help us better understand how these devices work. Example 3 Examining an Advertisement An advertisement for Top Power nutritional supplements makes the following claim: “If you use Top Power, then you will not feel tired and you will get a solid workout.” Translate the statement into symbolic form and construct a truth table. Solution Let p: You use Top Power. q: You will feel tired. r: You will get a solid workout. In symbolic form, the claim is → ∼ ∧ p q r ( ). This symbolic statement is identical to the statement in Table 3.18. Thus, the truth tables are the same. Column 3 represents the truth value of ∼ ∧ q r ( ), which corresponds to the statement “You will not feel tired and you will get a solid workout.” Note that column 3 is true in cases (rows) 3 and 7. In case 3, since p is true, you used Top Power. In case 7, however, since p is false, you did not use Top Power. From case 7 we can conclude that it is possible for you to not feel tired and for you to get a solid workout without using Top Power. 7 Now try Exercise 25 Table 3.19 p q p( → q) ∧ (q → p) T T T T T T T T T T F T F F F F T T F T F T T F T F F F F F T F T F T F 1 3 2 7 4 6 5 Table 3.20 Biconditional p q ↔p q T T T T F F F T F F F T From Table 3.20 we see that the biconditional statement is true when the antecedent and the consequent have the same truth value and false when the antecedent and consequent have different truth values. Holbox/Shutterstock
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