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2. Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof. a. There is no integer that is both even and odd. b. For all integers x, y, z with y + 0 and z + 0, if x is divisible by y and y is divisible by z, then x is divisible by z. | z. c. For all integers x, y, z with z + 0, if xy is not divisible by z, then x is not divisible by z.

2. Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof. a. There is no integer that is both even and odd. b. For all integers x, y, z with y + 0 and z + 0, if x is divisible by y and y is divisible by z, then x is divisible by z. | z. c. For all integers x, y, z with z + 0, if xy is not divisible by z, then x is not divisible by z.

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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Do Part  A, B, C please.

Part II: Constructing proofs You must write down all proofs in acceptable mathematical language: mark the beginning and end of the proof, state every assumption, define every variable, give a justification for every assertion (e.g., by definition of...), and use complete, grammatically correct sentences. See lecture slides for examples. Definitions: An integer n is even if and only if there exists an integer k such that n = 2k. An integer n is odd if and only if there exists an integer k such that n = 2k + 1. Two integers have the same parity when they are both even or when they are both odd. Two integers have opposite parity when one is even and the other one is odd. • An integer n is divisible by an integer d with d + 0, denoted d | n, if and only if there exists an integer k such that n = dk. A real number r is rational if and only if there exist integers a and b with b + 0 such that r = a/b. • For any real number x, the absolute value of x, denoted |x|, is defined as follows: |x| = { x if x > 0 l-x if x < 0 2. Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof. a. There is no integer that is both even and odd. b. For all integers x, y, z with y + 0 and z + 0, if x is divisible by y and y is divisible by z, then x is divisible by z. c. For all integers x, y, z with z + 0, if xy is not divisible by z, then x is not divisible by z. d. The sum of any rational number and any irrational number is irrational. e. Any two consecutive integers have opposite parity. f. For any real number x, |x –- 7|+x2 7.
Transcribed Image Text:Part II: Constructing proofs You must write down all proofs in acceptable mathematical language: mark the beginning and end of the proof, state every assumption, define every variable, give a justification for every assertion (e.g., by definition of...), and use complete, grammatically correct sentences. See lecture slides for examples. Definitions: An integer n is even if and only if there exists an integer k such that n = 2k. An integer n is odd if and only if there exists an integer k such that n = 2k + 1. Two integers have the same parity when they are both even or when they are both odd. Two integers have opposite parity when one is even and the other one is odd. • An integer n is divisible by an integer d with d + 0, denoted d | n, if and only if there exists an integer k such that n = dk. A real number r is rational if and only if there exist integers a and b with b + 0 such that r = a/b. • For any real number x, the absolute value of x, denoted |x|, is defined as follows: |x| = { x if x > 0 l-x if x < 0 2. Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof. a. There is no integer that is both even and odd. b. For all integers x, y, z with y + 0 and z + 0, if x is divisible by y and y is divisible by z, then x is divisible by z. c. For all integers x, y, z with z + 0, if xy is not divisible by z, then x is not divisible by z. d. The sum of any rational number and any irrational number is irrational. e. Any two consecutive integers have opposite parity. f. For any real number x, |x –- 7|+x2 7.
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