3.3 Truth Tables for the Conditional and Biconditional 127 When every truth value in the answer column of the truth table is true, the statement is a tautology. Example 8 All Falses, a Self-Contradiction Construct a truth table for the statement ↔ ∧ ↔∼ p q p q ( ) ( ). Solution See Table 3.22. In this example, the truth values are false in each case of column 5. This statement is an example of a self-contradiction or a logically false statement. Table 3.22 p q p q ( ) ↔ ∧ p( ↔ q) ∼ T T T F T F F T F F F T T T F T F F F T F F F T F F F T 1 5 2 4 3 7 Now try Exercise 33 Definition: Tautology A tautology is a compound statement that is always true. Example 9 All Trues, a Tautology Construct a truth table for the statement p q p r ( ) ( ). ∧ → ∨ Solution The answer is given in column 3 of Table 3.23. The truth values are true in every case. Thus, the statement is an example of a tautology or a logically true statement. Table 3.23 p q r p q ( ) ∧ → p r ( ) ∨ T T T T T T T T F T T T T F T F T T T F F F T T F T T F T T F T F F T F F F T F T T F F F F T F 1 3 2 Now try Exercise 31 7 The conditional statement p q p r ( ) ( ) ∧ → ∨ is a tautology. Conditional statements that are tautologies are called implications. In Example 9, we can say that p q ∧ implies p r. ∨
RkJQdWJsaXNoZXIy NjM5ODQ=