(d) Give an example of an equivalence relation on the set {1, 2, 3} with exactly two equivalence classes. (e) Given S is the set of integers (2, 3, 4, 6, 7, 9). Let R be a relation defined on S by the following condition such that, for all z, y E S, xRy if 3|(x + y) which means 3 divides (z – y). i. Draw the digraph of R. ii. Say with reason whether or not R is • reflexive; • symmetric; • anti-symmetric; • transitive. In the cases where the given property does not hold, provide a counter example to justify this.
(d) Give an example of an equivalence relation on the set {1, 2, 3} with exactly two equivalence classes. (e) Given S is the set of integers (2, 3, 4, 6, 7, 9). Let R be a relation defined on S by the following condition such that, for all z, y E S, xRy if 3|(x + y) which means 3 divides (z – y). i. Draw the digraph of R. ii. Say with reason whether or not R is • reflexive; • symmetric; • anti-symmetric; • transitive. In the cases where the given property does not hold, provide a counter example to justify this.
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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Question

Transcribed Image Text:(d) Give an example of an equivalence relation on the set {1,2, 3} with exactly
two equivalence classes.
(e) Given S is the set of integers (2, 3, 4, 6, 7, 9.. Let R be a relation defined
on S by the following condition such that,
for all r, y e S, xRy if 3|(x + y) which means 3 divides (x – y).
i. Draw the digraph of R.
ii. Say with reason whether or not R is
• reflexive;
• symmetric;
• anti-symmetric;
• transitive.
In the cases where the given property does not hold, provide a
counter example to justify this.
i. is Ra partial order? Explain your answer.
iv. is R an equivalence relation?
(f) Letf : A → B and g : B → C be functions. Prove that if g o f is one-to-one,
then f is one-to-one.
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