114 CHAPTER 3 Logic We have learned that a truth table with one simple statement has two cases, a truth table with two simple statements has four cases, and a truth table with three simple statements has eight cases. In general, the number of distinct cases in a truth table with n distinct simple statements is 2 .n The compound statement p q r s ( ) ( ) ∨ ∨ ∧ ∼ has four simple statements, p, q, r, s. Thus, a truth table for this compound statement would have 2 , 4 or 16, distinct cases. Place values under p, column 1, and ∼q, column 2. Then determine the conjunction of columns 1 and 2 to obtain column 3. Place the values of r in column 4. To obtain the answer, column 5, use columns 3 and 4 and the information for the disjunction contained in Table 3.5 on page 109. b) We are given the following: p q r : Joaquin is working late—false. : Joaquin is fishing—true. : Joaquin is sleeping—true. We need to determine the truth value of the following case: false, true, true. In case 5 of the truth table, p, q, and r are F, T, and T, respectively. Therefore, under these conditions, the original compound statement is true (as circled in the table). 7 Now try Exercise 13 TECHNOLOGY TIP Truth Table Builder Apps There are several apps available for use on most smartphones and tablets that can assist in the building of truth tables. Most of these apps can also be used to test your ability to complete truth tables accurately. Although these apps can be helpful as you learn how to build truth tables, be careful to completely understand the instructions and notation used before utilizing such an app. For example, the plus sign, +, is often used to represent “and”; a vertical bar (found on most computer keyboards above the backward slash), I, is often used to represent “or”; and the apostrophe,’, is often used to represent “not.” Finally, as always, check with your instructor prior to using any apps to complete required work for your course. Alternate Method for Constructing Truth Tables We now present an alternate method for constructing truth tables. We will use the alternate method to construct truth tables for the same statements we analyzed in Examples 1, 2, and 3. Example 6 Use the Alternate Method to Construct a Truth Table Construct a truth table for p q. ∧ ∼ Solution We begin by constructing the first two columns of a truth table with four cases, as shown in Table 3.12(a). We will add additional columns to Table 3.12(a) to develop our answer column. Since we wish to determine the truth table for the compound statement p q, ∧ ∼ we need to be able to compare the truth values for p with the truth values for q. ∼ Table 3.12(a) already has a column showing the truth values for p. We next add a column showing the truth values for q, ∼ as shown in Table 3.12(b). Recall that the values of q∼ are the opposite of those for q. Finally, we add the answer column for the compound statement p q , ∧ ∼ as shown in Table 3.12(c). To determine the truth values for the p q ∧ ∼ column, use the p column and the q∼ column, and the conjunction table, Table 3.4, on page 108.
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