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College Algebra
College Algebra
10th Edition
ISBN:9781337282291
Author:Ron Larson
Publisher:Ron Larson
Chapter8: Sequences, Series,and Probability
Section8.6: Counting Principles
Problem 22E: License Plate Numbers In a certain state, each automobile license plate number consists of two...
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• Prove that the square of any odd integer is odd. An integer n is odd iff there exists an integer k such that n = 2k + 1. Start: Suppose n is an odd integer. n = 2r + 1 FOR SOME INTEGER r n²= (2r + 1)²= 4r*+4r + 1 = 2(2r²+ 2r) + 1 %3D LET S= 2r²+ 2r , s iS AN INTEGER %3D n°: 2s + 1 FOR SOME INTEGERS THEREFORE, n² is ODD End: n? is odd.
Transcribed Image Text:• Prove that the square of any odd integer is odd. An integer n is odd iff there exists an integer k such that n = 2k + 1. Start: Suppose n is an odd integer. n = 2r + 1 FOR SOME INTEGER r n²= (2r + 1)²= 4r*+4r + 1 = 2(2r²+ 2r) + 1 %3D LET S= 2r²+ 2r , s iS AN INTEGER %3D n°: 2s + 1 FOR SOME INTEGERS THEREFORE, n² is ODD End: n? is odd.
Additional Topics: Proving biconditional statements To prove that a biconditional statement of the form p e q is true, you must prove that p → q and q → p are both true. For example, to prove that for any integer n, n is odd if and only if n? is odd, you must prove that (1) if n is odd, then n? is odd (see Example 2 in Lecture Slides 08), and (2) if n² is odd, then n is odd (see Exercise 10.2.1 in Lecture Slides 10). 3. Prove that for any positive integer n, n is even if and only if 7n + 4 is even. Indicate which proof methods you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.
Transcribed Image Text:Additional Topics: Proving biconditional statements To prove that a biconditional statement of the form p e q is true, you must prove that p → q and q → p are both true. For example, to prove that for any integer n, n is odd if and only if n? is odd, you must prove that (1) if n is odd, then n? is odd (see Example 2 in Lecture Slides 08), and (2) if n² is odd, then n is odd (see Exercise 10.2.1 in Lecture Slides 10). 3. Prove that for any positive integer n, n is even if and only if 7n + 4 is even. Indicate which proof methods you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.
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