(c) Consider the equivalence relation R on the set X = {1,2,3,4,5,6,7} given by R = {(1,1),(1,6), (2, 2), (2,5), (2,7), (3, 3), (3,4), (4,3), (4,4), (5,2), (5,5), (5,7), (6, 1), (6,6), (7,2), (7,5), (7,7)} Write out the equivalence classes of R and find the partition of X induced by R (you are not required to check that R is an equivalence relation). (d) Prove the following statement. Let R be an equivalence relation on X and let x, y € X. If (x, y) = R, then the equivalence classes R(x) and R(y) are equal. In symbols, (x,y) ≤R ⇒ R(x) = R(y).
(c) Consider the equivalence relation R on the set X = {1,2,3,4,5,6,7} given by R = {(1,1),(1,6), (2, 2), (2,5), (2,7), (3, 3), (3,4), (4,3), (4,4), (5,2), (5,5), (5,7), (6, 1), (6,6), (7,2), (7,5), (7,7)} Write out the equivalence classes of R and find the partition of X induced by R (you are not required to check that R is an equivalence relation). (d) Prove the following statement. Let R be an equivalence relation on X and let x, y € X. If (x, y) = R, then the equivalence classes R(x) and R(y) are equal. In symbols, (x,y) ≤R ⇒ R(x) = R(y).
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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