116 CHAPTER 3 Logic Now try Exercise 25 Finally, we add the answer column for the compound statement, ∼ ∼ ∨ q p ( ), as shown in Table 3.15. To determine the truth values for the ∼ ∼ ∨ q p ( ) column, take the opposite values of those shown in the ∼ ∨ q p column in Table 3.14(c). Note that the answer column of Table 3.15 is the same as the answer column of Table 3.8 on page 112. Table 3.14 (a) p q T T T F F T 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 T T F T T F T F F T F T T Use these columns to determine the ∼ ∨ q p column. ∼ ∨ q p column Table 3.15 p q q∼ q p ∼ ∨ ∼ ∼ ∨ ( ) q p T T F T F T F T T F F T F F T F F T T F Take the opposite of this column to get the answer column. Answer column 7 We have demonstrated two methods for constructing truth tables. Unless your instructor indicates otherwise, you may use either method. Both methods, if done correctly, will lead to the correct answer. In the remainder of this chapter, we will demonstrate the construction of truth tables using only the first method. Determine Truth Values Without Constructing a Truth Table When we construct a truth table, we determine the truth values of a compound statement for every possible case. If we want to determine the truth value of the compound statement for any specific case when we know the truth values of the simple statements, we do not have to develop the entire table. For example, to determine the truth value for the statement + = + = 2 3 5 and 1 1 3 we let p be + = 2 3 5 and q be + = 1 1 3. Now we can write the compound statement as ∧ p q. We know that p is a true statement and q is a false statement. Thus, we can substitute T for p and F for q and evaluate the statement: ∧ ∧ p q T F F Therefore, the compound statement + = 2 3 5 and + = 1 1 3 is a false statement. In the remaining examples in this section, we will determine the truth values of compound statements without constructing a truth table. Exploring Truth Values for Compound Statements Example 9 Determine the Truth Value of Compound Statements Determine the truth value of each simple statement. Then, using these truth values, determine the truth value of the compound statement. a) 1 million is greater than or equal to 1 billion. b) Montreal is in Canada or Beijing is in Mexico, but Santiago is not in Japan.
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