By Abramowitz M., Stegun I.A. (eds.)
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Juxtapose those octal digits. As an example consider converting 11010111012 to octal notation. Grouping the binary digits in sets of three and adding two leading zeros gives 001 101 011 101. To convert each group into an octal digit, note that 0012 = 18 1012 = 58 0112 = 38 1012 = 58 So the octal version of the integer should be 15358. To check this result, observe that 1 101011012= ( 2 9 +1 28+1. 2 6 +l. 83 + 5 82 + 3 8 + 5)1o = 861l( also. 2 +)O=8611o Chapter 2: The Logic of Quantified Statements Ability to use the logic of quantified statements correctly is necessary for doing mathematics because mathematics is, in a very broad sense, about quantity.
Hence, by substitution, n3 - m 3 is even [as was to be shown]. 47. Counterexample: Let n = 2. Then n is prime but 48. Counterexample: Let m = 3. Then 49. Counterexample: Let number. n = 11. m2 - Then n 2 -n (-I )n (_1)2 = 1 7 -1. 4 = 9 -4 = 5, which is not composite. + 11 112 -11 + 11 = 112, which is not a prime 46 Solutions for Exercises: Elementary Number Theory and Methods of Proof 51. Counterexample: The number 28 cannot be expressed as a sum of three or fewer perfect squares. The only perfect squares that could be used to add up to 28 are those that are smaller than 28: 1, 4, 9, 16, and 25.
If a person is a true poet, then he can stir the human heart. 4. If a writer can stir the human heart, then that writer understands human nature. 1. If a writer understands human nature, then that writer is clever. Shakespeare was clever. 35. The law of universal modus tollens says that the following form of argument is valid: Vx in D, if P(x) then Q(x). Q(c) for a particular c in D. P(c). m *- Major premise minor premise 40 Solutions for Exercises: The Logic of Quantified Statements Proof of Validity: Suppose the major and minor premises of the above argument form are both true.