110 CHAPTER 3 Logic Constructing Truth Tables for Statements Involving Negations, Conjunctions, and Disjunctions We will now construct additional truth tables for statements involving the negation, conjunction, and disjunction. We summarize these compound statements below. Learning Catalytics Keyword: Angel-SOM-3.2 (See Preface for additional details.) Negation, Conjunction, and Disjunction • Negation, ,p∼ is read “not p.” If p is true, then p∼ is false; if p is false, then p∼ is true. In other words, p∼ will always have the opposite truth value of p. • Conjunction, ∧ , p q is read “p and q.” ∧ p q is true only when both p and q are true. • Disjunction, ∨ , p q is read “p or q.” ∨ p q is true when either p is true, q is true, or both p and q are true. In other words, ∨ p q is false only when both p and q are false. We will discuss two methods for constructing truth tables. Although the two methods produce tables that will look different, the answer columns will be the same regardless of which method you use. Example 1 Construct a Truth Table Construct a truth table for p q. ∧ ∼ Solution Because there are two statements, p and q, construct a truth table with four cases; see Table 3.6(a). Then write the truth values under the p in the compound statement and label this column 1, as in Table 3.6(b). Copy these truth values directly from the p column on the left. Write the corresponding truth values under the q in the compound statement and call this column 2, as in Table 3.6(c). Copy the truth values for column 2 directly from the q column on the left. Now find the truth values of q∼ by negating the truth values in column 2 and call this (a) p q p q ∧ ∼ Case 1 T T Case 2 T F Case 3 F T Case 4 F F (b) p q p q ∧ ∼ T T T T F T F T F F F F 1 (c) p q p q ∧ ∼ T T T T T F T F F T F T F F F F 1 2 (d) p q p ∧ ∼ q T T T F T T F T T F F T F F T F F F T F 1 3 2 (e) p q p ∧ ∼ q T T T F F T T F T T T F F T F F F T F F F F T F 1 4 3 2 Answer column Table 3.6
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