2.1.2 Conditional Statements (continued) Example 2 Writing a Conditional Statement
In each of these three examples, the given information must be used to write a conditional statement. So, first identify the two parts from the given information, then determine which part depends on the other, and then phrase the two parts in the “if/then” form. In the first example, the two parts are “a cat is an animal” and “that can climb a tree.” The first part is the hypothesis and the second part is the conclusion. The second example may appear to have only one part. But, the two parts are “an angle is a right angle” and “an angle measures 90°.” The information in the third example is given in a Venn diagram. The Venn diagram shows two parts, the group of vegetables and the group of carrots. When writing a conditional using a Venn diagram, the hypothesis is always gathered from the information given in the inner oval and the conclusion is gathered from the information given in the outer oval.
Example 3 Analyzing the Truth Value of a Conditional Statement A conditional statement has a truth value of either true (T) or false (F), which depends upon the truth value of the hypothesis and the conclusion. A conditional statement’s truth value is false only when the hypothesis is true and the conclusion is false. Otherwise, the conditional’s truth value is true. So, the conditional statement’s truth value is true in all other cases, including when the hypothesis is false. If the hypothesis is false, the conditional is true and the truth value of the conclusion is irrelevant. Determine the truth value of a conditional by examining the truth value of the hypothesis and the conclusion. Showing that a conditional statement is false requires only one counterexample, which is an example where the hypothesis is true but the conclusion is false.
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