1. Consider the result: For all integers x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. (a) Write a paragraph proof of this result using the proof of contrapositive technique. (b) Write a paragraph proof of this result using the proof by contradiction technique. 2. Construct a proof by contradiction to prove the result: No odd integer can be expressed as the sum of three even integers. 3. Suppose that you would like to prove the following implication: "For all numbers n, if n is prime then n is solitary." Write out the assumption you would state at the beginning of the argument if you were to prove the statement, (a) Directly (b) By contrapositive (c) By contradiction Note: You do not need to actually prove the result (since you do not know what solitary means).

1. Consider the result: For all integers x and y, if x + y ≥ 9, then either x ≥ 5 or y ≥ 5. (a) Write a paragraph proof of this result using the proof of contrapositive technique. (b) Write a paragraph proof of this result using the proof by contradiction technique. 2. Construct a proof by contradiction to prove the result: No odd integer can be expressed as the sum of three even integers. 3. Suppose that you would like to prove the following implication: "For all numbers n, if n is prime then n is solitary." Write out the assumption you would state at the beginning of the argument if you were to prove the statement, (a) Directly (b) By contrapositive (c) By contradiction Note: You do not need to actually prove the result (since you do not know what solitary means).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...

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