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Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a +b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a +b must be odd.

Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a +b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a +b must be odd.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. O Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a + b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a + b must be odd.
Transcribed Image Text:Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. O Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a + b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a + b must be odd.
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