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(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!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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a,b,c please using pigeonhole principle
(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 ZZ x ZZ+ (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 (22,92) in S such that ₁+2 and y₁ +92 are both even? Explain. (c) Generalize (b) (and hance (a)) to k-tuples of positive integers. Justify your reasoning!
Transcribed Image Text:(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 ZZ x ZZ+ (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 (22,92) in S such that ₁+2 and y₁ +92 are both even? Explain. (c) Generalize (b) (and hance (a)) to k-tuples of positive integers. Justify your reasoning!
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