2.1.2 Conditional Statements (continued)
The hypothesis in the first example is true because a month can be April. The conclusion is true because the month that follows April is always May; there is no counterexample. So, since the hypothesis and the conclusion are true, the conditional is also true. For the second example, consider the truth value of the hypothesis. The hypothesis is true because a quadrilateral can have four 90° angles. Now consider the conclusion. Is it always true that a quadrilateral with four 90° angles is a square? A square does have four 90° angles, so the conclusion can be true, but it is not always true. A counterexample is the case of a rectangle because a rectangle is a quadrilateral with four 90° angles and is not a square. Thus, because the hypothesis is true and the conclusion is false, the conditional is false. In the third example, again begin by considering the truth value of the hypothesis. Is the statement “121 is a prime number” true? No, the hypothesis is false because 121 is not a prime number (since 11(11) = 121). So, the conditional is true because the hypothesis is false. There is no need to consider the truth value of the conclusion in this case.
Example 4 Writing Related Conditionals Remember, a conditional is a statement that can be written in the form “if p , then q ,” where p is the hypothesis and q is the conclusion. Related conditionals are statements based on a given conditional. In a related conditional, the hypothesis and conclusion of the given conditional are either reversed, negated, or both reversed and negated. The negation of a statement p is “not p ,” written as ~ p .
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