2.1.4 Biconditional Statements and Definitions (continued)
In this example, p is defined to be “a solution is an acid” and q is defined to be “it has a pH less than 7.”
Example 2 Writing a Biconditional Statement To write the converse of a given conditional, just reverse the hypothesis and the conclusion. To write the biconditional from a given conditional, remove “if” and replace “then” with “if and only if.”.
When writing a related conditional or biconditional from a given conditional, it is acceptable to replace pronouns such as “it” or “they.” The hypothesis of the conditional given in the first example is “10 = 7 x − 4” and the conclusion is “ x = 2.” Switch the hypothesis and the conclusion to write the converse. Leave the hypothesis and the conclusion as they are in the original conditional to write the biconditional, but remove “if” and replace “then” with “if and only if.” In the conditional given in the second example, the hypothesis is “a triangle has three equal sides” and the conclusion is “it is equilateral.”
Example 3 Analyzing the Truth Value of a Biconditional Statement Remember, a conditional has a truth value of either true or false. A conditional can be proved to be false by providing a single counterexample to the conditional where the hypothesis is true, but the conclusion is false. A biconditional statement is true only when the related conditional and its converse are true. If either the related conditional or its converse are false, then the biconditional statement is also false.
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