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Suppose R, S are equivalence relations on set A. Prove or disprove that (a) R-¹nS is also an equivalence relation on A; (b) RoS is also an equivalence relation on A. ((Hint: Not symmetric. An example. Let A = {a,b,c) and S = {(a, a), (b, b), (c, c), (a, b), (b, a)} and S= {(a, a), (b, b), (c, c), (b, c), (c, b)}. Then RoS={(a, a), (b, b), (c, c), (a, b), (b, a), (c, b), (a, c), (b, c), (c, b)} does not contain (c, a).

Suppose R, S are equivalence relations on set A. Prove or disprove that (a) R-¹nS is also an equivalence relation on A; (b) RoS is also an equivalence relation on A. ((Hint: Not symmetric. An example. Let A = {a,b,c) and S = {(a, a), (b, b), (c, c), (a, b), (b, a)} and S= {(a, a), (b, b), (c, c), (b, c), (c, b)}. Then RoS={(a, a), (b, b), (c, c), (a, b), (b, a), (c, b), (a, c), (b, c), (c, b)} does not contain (c, a).

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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Prove or disprove the following statements

Suppose R, S are equivalence relations on set A. Prove or disprove that (a) R-¹nS is also an equivalence relation on A; (b) RoS is also an equivalence relation on A. ((Hint: Not symmetric. An example. Let A = {a,b,c) and S = {(a, a), (b, b), (c, c), (a, b), (b, a)} and S= {(a, a), (b, b), (c, c), (b, c), (c, b)}. Then RoS= {(a, a), (b, b), (c, c), (a, b), (b, a), (c, b), (a, c), (b, c), (c, b)} does not contain (e, a).
Transcribed Image Text:Suppose R, S are equivalence relations on set A. Prove or disprove that (a) R-¹nS is also an equivalence relation on A; (b) RoS is also an equivalence relation on A. ((Hint: Not symmetric. An example. Let A = {a,b,c) and S = {(a, a), (b, b), (c, c), (a, b), (b, a)} and S= {(a, a), (b, b), (c, c), (b, c), (c, b)}. Then RoS= {(a, a), (b, b), (c, c), (a, b), (b, a), (c, b), (a, c), (b, c), (c, b)} does not contain (e, a).
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