2.1.4 Biconditional Statements and Definitions (continued)
The given conditional is true only if the related conditional and its converse are true. So, write the conditional and its converse using the hypothesis p and the conclusion q from the given biconditional. Then, determine the truth value of each. If a counterexample can be found for either the conditional or the converse, then the biconditional is false. A counterexample can be found for the converse since a rectangle with area 30 does exist where its length and width are not 10 and 3, respectively. Specifically, a rectangle with length 15 and width 2 has area 30, but does not have length 10 and width 3. Therefore, since the converse is false, the biconditional is false. In this example, use the definition of acute angles to determine the truth value of the conditional and its converse. An acute angle is defined to be an angle with measure that is less than 90°.
Example 4 Writing Definitions as Biconditional Statements
A definition is a statement that describes an object and can be written as a true biconditional. In the first example, the definition can be written as a biconditional by simply replacing “is” with “if and only if” and rephrasing the definition slightly. The definition in the second example can be written as a biconditional by replacing “that” with “if and only if” and rephrasing the definition.
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