3.2 Truth Tables for Negation, Conjunction, and Disjunction 113 So far, all the truth tables we have constructed have contained at most two simple statements. Now we will explain how to construct a truth table that consists of three simple statements, such as ∧ ∧ p q r ( ) . When a compound statement consists of three simple statements, there are eight different true–false possibilities, as illustrated in Table 3.10. To begin such a truth table, write four T’s and four F’s in the column under p. Under the second statement, q, pairs of T’s alternate with pairs of F’s. Under the third statement, r, T alternates with F. This technique is not the only way of listing the cases, but it ensures that each case is unique and that no cases are omitted. Example 4 Use the General Procedure to Construct a Truth Table Construct a truth table for the statement p q p ( ) . ∧ ∼ ∨ ∼ Solution This statement is a disjunction, so the answer will be under the disjunction symbol, ∨. To begin, we complete the columns under p and ∼ q within the parentheses and call these columns 1 and 2, respectively (see Table 3.9). Next, complete the column under the conjunction symbol, ∧, using the truth values in columns 1 and 2, and call this column 3. Then, complete the column under p∼ and call this column 4. To complete the answer, column 5, use the definition of disjunction and the truth values in columns 3 and 4. 7 Now try Exercise 35 Table 3.9 p q p( ∧ q) ∼ ∨ p∼ T T T F F F F T F T T T T F F T F F F T T F F F F T T T 1 3 2 5 4 Table 3.10 p q r Case 1 T T T Case 2 T T F Case 3 T F T Case 4 T F F Case 5 F T T Case 6 F T F Case 7 F F T Case 8 F F F Example 5 Use the General Procedure to Construct a Truth Table with Eight Cases a) Construct a truth table for the statement “Joaquin is working late and he is not fishing, or he is sleeping.” b) Suppose that “Joaquin is working late” is a false statement, that “Joaquin is fishing” is a true statement, and that “Joaquin is sleeping” is a true statement. Is the compound statement in part (a) true or false? Solution a) First we translate the statement into symbolic form. Let p q r : Joaquin is working late. : Joaquin is fishing. : Joaquin is sleeping. In symbolic form, the statement is p q r ( ) . ∧ ∼ ∨ Since the statement is composed of three simple statements, there are eight cases. Begin by listing the eight cases in the three left-hand columns; see Table 3.11. By examining the statement, you can see that it is a disjunction. Therefore, the answer will be in the ∨ column. Fill out the truth table by working in parentheses first. Table 3.11 p q r p( ∧ q) ∼ ∨ r T T T T F F T T T T F T F F F F T F T T T T T T T F F T T T T F F T T F F F T T F T F F F F F F F F T F F T T T F F F F F T F F 1 3 2 5 4
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