3.2 Truth Tables for Negation, Conjunction, and Disjunction 111 Row 1: T F is F. Row 2: T T is T. Row 3: F F is F. Row 4: F T is F. ∧ ∧ ∧ ∧ The answer is always the last column completed . The columns labeled 1, 2, and 3 assist in arriving at the answer labeled column 4. 7 Now try Exercise 7 The statement p q ∧ ∼ in Example 1 actually means p q ( ). ∧ ∼ In the future, instead of listing a column for q and a separate column for its negation, we will make one column for q, ∼ which will have the opposite values of those in the q column on the left. Similarly, when we evaluate p, ∼ we will use the opposite values of those in the p column on the left. This procedure is illustrated in Example 2. In Example 1, we spoke about cases and also columns . Consider Table 3.6(e). This table has four cases indicated by the four different rows of the two left-hand (unnumbered) columns. The four cases are TT, TF, FT, and FF. In every truth table with two letters, we list the four cases (the first two columns) first. Then we complete the remaining columns in the truth table. In Table 3.6(e), after completing the two left-hand columns, we complete the remaining columns in the order indicated by the numbers below the columns. We will continue to place numbers below the columns to show the order in which the columns are completed. In discussion of the truth table in Example 2, and all following truth tables, if we say column 1, it means the column labeled 1. Column 2 will mean the column labeled 2, and so on. column 3, as in Table 3.6(d). Use the conjunction table, Table 3.4, and the entries in the columns labeled 1 and 3 to complete the column labeled 4, as in Table 3.6(e). The results in column 4 are obtained as follows: Timely Tip When constructing truth tables it is very important to keep your entries in neat columns and rows. If you are using lined paper, put only one row of the table on each line. If you are not using lined paper, using a straightedge may help you correctly enter the information into the truth table’s rows and columns. Example 2 Construct and Interpret a Truth Table a) Construct a truth table for the following statement: The lamp is not a Himalayan salt lamp and the crystal is not amethyst. b) Under what conditions will the compound statement be true? c) Suppose “The lamp is a Himalayan salt lamp” is a false statement and “The crystal is amethyst” is a true statement. Is the compound statement given in part (a) true or false? Solution a) First, write the simple statements in symbolic form by using simple statements that are not negations. Let p q : The lamp is a Himalayan salt lamp. : The crystal is amethyst. m A Himalayan salt lamp Therefore, the compound statement may be written p q. ∼ ∧ ∼ Now construct a truth table with four cases, as shown in Table 3.7. Table 3.7 p q p∼ ∧ q∼ T T F F F T F F F T F T T F F F F T T T 1 3 2 Liudmilachernetska/123RF
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