2.1.3 Using Deductive Reasoning to Verify Conjectures (continued) Example 2 Verifying Conjectures by Using the Law of Detachment One case in which a conditional statement is true is when the hypothesis is true and the conclusion is true. If the hypothesis of a conditional statement is true and the conclusion is false, then the conditional statement cannot be true. Therefore, it can be deduced that when it is known that a conditional statement is true and it is known that the hypothesis is true, it must be the case that the conclusion is also true. The Law of Detachment summarizes this concept. According to the Law of Detachment, if p → q (a conditional) is a true statement and p (its hypothesis) is true, then q (its conclusion) is true.
In order for a conjecture to be valid by the Law of Detachment, a conditional and its hypothesis must be true and the conjecture must be equivalent to the conditional’s conclusion. In this example, the conditional is given to be true and the given statement matches the hypothesis of the given conditional. The conjecture is equivalent to the conditional’s conclusion, so the conjecture holds by the Law of Detachment. In this example, the given conditional is true, but the given statement matches the conclusion of the given conditional, not the conditional’s hypothesis. So, the conjecture is not valid by the Law of Detachment.
Example 3 Verifying Conjectures by Using the Law of Syllogism The Law of Detachment is one example of deductive reasoning. Another example is the Law of Syllogism, which states that if p → q and q → r are true statements, then p → r is a true statement. In other words, if two given conditionals are true, and the same statement is used for the conclusion of one conditional and for the hypothesis of the other conditional, then the conditional formed by the non-matching hypothesis and conclusion must also be true. Specifically, if the first conditional’s hypothesis matches the second conditional’s conclusion, then by the Law of Syllogism, the conditional formed from the hypothesis of the first conditional and the conclusion of the second conditional must be true.
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