3.2 Truth Tables for Negation, Conjunction, and Disjunction 115 Table 3.12 Now try Exercise 21 (a) p q Case 1 T T Case 2 T F Case 3 F T Case 4 F F (b) p q q∼ T T F T F T F T F F F T (c) p q q∼ q p ∧ ∼ T T F F T F T T F T F F F F T F Use these columns to determine the answer column. Answer column Note that the answer column of Table 3.12(c) is the same as the answer column of Table 3.6(e) on page 110. 7 Example 7 Use the Alternate Method to Construct a Truth Table Construct a truth table for p q. ∼ ∧ ∼ Solution Begin by constructing a truth table with four cases, as shown in Table 3.13(a). Since we wish to determine the truth table for the compound statement p q , ∼ ∧ ∼ we will add a column for p∼ and a column for q, ∼ as shown in Table 3.13(b). Now try Exercise 23 Table 3.13 (a) p q T T T F F T F F (b) p q p∼ q∼ T T F F T F F T F T T F F F T T (c) p q p∼ q∼ p q ∼ ∧ ∼ T T F F F T F F T F F T T F F F F T T T Finally, we add our answer column for the compound statement p q. ∼ ∧ ∼ 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. Note that the answer column of Table 3.13(c) is the same as the answer column of Table 3.7 on page 111. 7 Use these columns to determine the answer column. Answer column Example 8 Use the Alternate Method to Construct a Truth Table Construct a truth table for q p ( ). ∼ ∼ ∨ Solution Begin by constructing a truth table with four cases, as shown in Table 3.14(a). To complete the truth table, we will work within parentheses first. Thus, we next add a column for q, ∼ as shown in Table 3.14(b). We then will construct a column for the expression within parentheses q p ∼ ∨ by using the q∼ column and the p column, and the disjunction table, Table 3.5, on page 109.
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