3.4 Equivalent Statements 139 Since the contrapositive statement is always equivalent to the original conditional statement, in Example 11(d) we should have expected the answer to be a true statement because the original conditional statement was also a true statement. b) The converse of the conditional statement in symbols is q p. → Therefore, in words the converse is If the number is divisible by 5, then the number is divisible by 10. This statement is false. For example, 15 is divisible by 5, but 15 is not divisible by 10. c) The inverse of the conditional statement in symbols is p q ∼ →∼ Therefore, in words the inverse is If the number is not divisible by 10, then the number is not divisible by 5. This statement is false. For example, 25 is not divisible by 10, but 25 is divisible by 5. d) The contrapositive of the conditional statement in symbols is q p. ∼ →∼ Therefore, in words the contrapositive is If the number is not divisible by 5, then the number is not divisible by 10. This statement is true. Any number that is not divisible by 5 cannot be divisible by 10, since 5 is a divisor of 10. 7 Now try Exercise 45 Example 12 Use the Contrapositive Use the contrapositive to write a statement logically equivalent to “If you don’t eat your meat, then you can’t have any pudding.” Solution Let p q : You do eat your meat. : You can have any pudding. The given statement written symbolically is p q ∼ →∼ The contrapositive of the statement is q p → Therefore, an equivalent statement is “If you can have any pudding, then you do eat your meat.” 7 Now try Exercise 49 The contrapositive of the conditional is very important in mathematics. Consider the statement “If a2 is not a whole number, then a is not a whole number.” Is this statement true? You may find this question difficult to answer. Writing the statement’s contrapositive may enable you to answer the question. The contrapositive is “If a is a whole number, then a2 is a whole number.” Since the contrapositive is a true statement, the original statement must also be true. MATHEMATICS TODAY Fuzzy Logic Many modern computers work solely with two values, 1 or 1. This constraint makes it difficult for a computer to evaluate vague concepts that human beings deal with on a regular basis, such as bright, slow, and light . More and more computers are becoming more capable of handling such vague concepts, thanks to the introduction of fuzzy logic . Unlike the traditional computer logic, fuzzy logic is based on the assignment of a value between 0 and 1, inclusively, that can vary from setting to setting. For example, a camera using fuzzy logic may assign bright a value of 0.9 on a sunny day, 0.4 on a cloudy day, and 0.1 at night. Fuzzy logic also makes use of logical statements like those studied in this chapter. One such statement is “If X and Y, then Z.” For example, a computer chip in a camera programmed with fuzzy logic may use the rule “If the day is bright and the movement is slow , then let in less light .” Fuzzy logic is discussed further in Exercises 75 and 77. Why This Is Important Fuzzy logic is used in many different technologies that we encounter daily. In addition to the camera application described above, fuzzy logic is used to adjust commuter train speeds subject to many conditions. Such adjustments improve the safety and comfort of passengers. Fuzzy logic is also used in household appliances including dishwashers, laundry machines, and vacuum cleaners. Example 13 Which Are Equivalent? Determine which, if any, of the following statements are equivalent. You may use De Morgan’s laws, the fact that → ⇔∼ ∨ p q p q, information from the variations of the conditional, or truth tables. a) If you leave by 9 a.m. , then you will get to your destination on time. b) You do not leave by 9 a.m. or you will get to your destination on time. c) It is false that you will get to your destination on time or you did not leave by 9 a.m. d) If you do not get to your destination on time, then you did not leave by 9 a.m. Sint/Shutterstock
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