108 CHAPTER 3 Logic A truth table is a device used to determine when a compound statement is true or false. Five basic truth tables are used in constructing other truth tables. Three are discussed in this section (Tables 3.2, 3.4, and 3.5), and two are discussed in the next section. Section 3.5 uses truth tables in determining whether a logical argument is valid or invalid. Table 3.2 Negation p p∼ Case 1 T F Case 2 F T Table 3.4 Conjunction p q p q ∧ Case 1 T T T Case 2 T F F Case 3 F T F Case 4 F F F Table 3.3 p q Case 1 T T Case 2 T F Case 3 F T Case 4 F F Negation The first truth table is for negation. If p is a true statement, then the negation of p, “not p,” is a false statement. If p is a false statement, then “not p” is a true statement. For example, if the statement “The shirt is blue” is true, then the statement “The shirt is not blue” is false. These relationships are summarized in Table 3.2. For a simple statement, there are exactly two true–false cases, as shown. Conjunction If a compound statement consists of two simple statements p and q, there are four possible cases, as illustrated in Table 3.3. Consider the statement “The test is today and the test covers Chapter 5.” The simple statement “The test is today” has two possible truth values, true or false. The simple statement “The test covers Chapter 5” also has two possible truth values, true or false. Thus, for these two simple statements there are four distinct possible true–false arrangements. Whenever we construct a truth table for a compound statement that consists of two simple statements, we begin by listing the four true–false cases shown in Table 3.3. To illustrate the conjunction, consider the following situation. You have recently purchased a new house. To decorate it, you ordered a new carpet and new furniture from the same store. You explain to the salesperson that the carpet must be delivered before the furniture. The salesperson promises that the carpet will be delivered on Thursday and that the furniture will be delivered on Friday. To help determine whether the salesperson kept their promise, we assign letters to each simple statement. Let p be “The carpet will be delivered on Thursday” and q be “The furniture will be delivered on Friday.” The salesperson’s statement written in symbolic form is ∧ p q. There are four possible true–false situations to be considered (Table 3.4). CASE 1: p is true and q is true. The carpet is delivered on Thursday and the furniture is delivered on Friday. The salesperson has kept his promise and the compound statement is true. Thus, we put a T in the ∧ p q column. CASE 2: p is true and q is false. The carpet is delivered on Thursday but the furniture is not delivered on Friday. Since the furniture was not delivered as promised, the compound statement is false. Thus, we put an F in the ∧ p q column. CASE 3: p is false and q is true. The carpet is not delivered on Thursday but the furniture is delivered on Friday. Since the carpet was not delivered on Thursday as promised, the compound statement is false. Thus, we put an F in the ∧ p q column. CASE 4: p is false and q is false. The carpet is not delivered on Thursday and the furniture is not delivered on Friday. Since the carpet and furniture were not delivered as promised, the compound statement is false. Thus, we put an F in the ∧ p q column. Examining the four cases, we see that in only one case did the salesperson keep their promise: in case 1. Therefore, case 1 (TT) is true. In cases 2, 3, and 4, the salesperson did not keep their promise and the compound statement is false. As this
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