(a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y in S such that x + y is even. (b) Suppose S is a subset of 72 x 7Z* (ordered pairs of positive integers). What is the minimal value of 5 in order to guarantee the existence of distinct ordered pairs (r1, y1) and (12,92) in S such that ₁ + x2 and y₁ + y2 are both even? Explain. (c) Generalize (b) (and hance (a)) to k-tuples of positive integers. Justify your reasoning!

(a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y in S such that x + y is even. (b) Suppose S is a subset of 72 x 7Z* (ordered pairs of positive integers). What is the minimal value of 5 in order to guarantee the existence of distinct ordered pairs (r1, y1) and (12,92) in S such that ₁ + x2 and y₁ + y2 are both even? Explain. (c) Generalize (b) (and hance (a)) to k-tuples of positive integers. Justify your reasoning!

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 23E

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