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(a) Give complete definitions of the terms (i) Cartesian product of two sets; (ii) relation on a set; (iii) equivalence relation on a set.

(a) Give complete definitions of the terms (i) Cartesian product of two sets; (ii) relation on a set; (iii) equivalence relation on a set.

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) Give complete definitions of the terms (i) Cartesian product of two sets; (ii) relation on a set; (iii) equivalence relation on a set. (b) Write down examples of: (i) a relation which is transitive but not reflexive; (ii) an equivalence relation on {1, 2, 3, 4} with exactly three equivalence classes. (c) Let X and Z be any two sets, and f : X - → Z any function. Prove that {(x, y) = X²: f(x) = f(y)} is an equivalence relation on X.
Transcribed Image Text:(a) Give complete definitions of the terms (i) Cartesian product of two sets; (ii) relation on a set; (iii) equivalence relation on a set. (b) Write down examples of: (i) a relation which is transitive but not reflexive; (ii) an equivalence relation on {1, 2, 3, 4} with exactly three equivalence classes. (c) Let X and Z be any two sets, and f : X - → Z any function. Prove that {(x, y) = X²: f(x) = f(y)} is an equivalence relation on X.
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