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Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x S such that: (a) R1 and R2 are partial orders but R1 U R2 is not. (b) Rị and R2 are equivalence relations but R1U R2 is not. (c) R1 and R2 are partial orders and R1 n R2 is an equivalence relation. (d) each R; is not reflexive, not symmetric, but transitive, while R1U R2 = S x S.

Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x S such that: (a) R1 and R2 are partial orders but R1 U R2 is not. (b) Rị and R2 are equivalence relations but R1U R2 is not. (c) R1 and R2 are partial orders and R1 n R2 is an equivalence relation. (d) each R; is not reflexive, not symmetric, but transitive, while R1U R2 = S x S.

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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Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x S such that: (a) R1 and R2 are partial orders but R1 U R2 is not. (b) Rị and R2 are equivalence relations but R1U R2 is not. (c) Rị and R2 are partial orders and R1 n R2 is an equivalence relation. (d) each R; is not reflexive, not symmetric, but transitive, while R1U R2 = S x S.
Transcribed Image Text:Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x S such that: (a) R1 and R2 are partial orders but R1 U R2 is not. (b) Rị and R2 are equivalence relations but R1U R2 is not. (c) Rị and R2 are partial orders and R1 n R2 is an equivalence relation. (d) each R; is not reflexive, not symmetric, but transitive, while R1U R2 = S x S.
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