112 CHAPTER 3 Logic Fill in the column labeled 1 by negating the truth values under p on the far left. Fill in the column labeled 2 by negating the values under q in the second column from the left. Fill in the column labeled 3 by using the columns labeled 1 and 2 and the definition of conjunction. In the first row, to determine the entry for column 3, we use false for p∼ and false for q. ∼ Since false ∧ false is false (see case 4 of Table 3.4 on page 108), we place an F in column 3, row 1. In the second row, we use false for p∼ and true for q. ∼ Since false ∧ true is false (see case 3 of Table 3.4), we place an F in column 3, row 2. In the third row, we use true for p∼ and false for q. ∼ Since true ∧ false is false (see case 2 of Table 3.4), we place an F in column 3, row 3. In the fourth row, we use true for p∼ and true for q. ∼ Since true ∧ true is true (see case 1 of Table 3.4), we place a T in column 3, row 4. b) The compound statement in part (a) will be true only in case 4 from 3.7 (circled in blue) when both simple statements, p and q, are false—that is, when the lamp is not a Himalayan salt lamp and the crystal is not an amethyst. c) We are told that p, “The lamp is a Himalayan salt lamp,” is a false statement and that q, “The crystal is amethyst,” is a true statement. From 3.7, we can determine that when p is false and q is true, the compound statement, case 3 (circled in red), is false. 7 Now try Exercise 9 Table 3.8 p q ∼ q (∼ ∨ p) T T F F T T T F F T T T F T T F F F F F F T T F 4 1 3 2 Example 3 Truth Table with a Negation Construct a truth table for q p ( ). ∼ ∼ ∨ Solution First construct the standard truth table listing the four cases. Then work within parentheses. The order to be followed is indicated by the numbers below the columns (see Table 3.8). Under q, ∼ column 1, write the negation of the q column. Then, in column 2, copy the values from the p column. Next, complete the or column, column 3, using columns 1 and 2 and the truth table for the disjunction (see Table 3.5 on page 109). The or column is false only when both statements are false, as in case 3. Finally, negate the values in the or column, column 3, and place these negated values in column 4. By examining the truth table you can see that the compound statement q p ( ) ∼ ∼ ∨ is true only in case 3, that is, when p is false and q is true. 7 Now try Exercise 11 GENERAL PROCEDURE USED FOR CONSTRUCTING TRUTH TABLES 1. Study the compound statement and determine whether it is a negation, conjunction, disjunction, conditional, or biconditional statement, as was done in Section 3.1. The answer to the truth table will appear under ∼ if the statement is a negation, under ∧ if the statement is a conjunction, under ∨ if the statement is a disjunction, under → if the statement is a conditional, and under ↔ if the statement is a biconditional. 2. Complete the columns under the simple statements, p, q, r, and their negations, p q r , , . ∼ ∼ ∼ If there are nested parentheses (one pair of parentheses within another pair), work with the innermost pair first. 3. Complete the column under the connective within the parentheses, if present. You will use the truth values of the connective in determining the final answer in Step 5. 4. Complete the column under any remaining statements and their negations. 5. Complete the column under any remaining connectives. Recall that the answer will appear under the column determined in Step 1. If the statement is a conjunction, disjunction, conditional, or biconditional, you will obtain the truth values for the connective by using the last column completed on the left side and on the right side of the connective. If the statement is a negation, you will obtain the truth values by negating the truth values of the last column completed within the grouping symbols on the right side of the negation. Be sure to circle or highlight your answer column or number the columns in the order they were completed. PROCEDURE
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