Question

on c

Answer 1

First of all note that, if 'S' is a set and ~ is a relation on S, then relation ~ is an equivalence relation if it is reflexive, symmetric and transitive. 1) here given ~ is an equivalence relation on the set s and x, y elementof s, with x ~ y To prove [x]_~ = [y]_~ Let z elementof [x]_~ then z ~ x --- (1) Also by hypothesis, we have x ~ y --- (2) so from (1), (2), we have z ~ y (Transitive property of ~) = > z elementof [y]_~ also, we see z elementof [x]_~ = > z elementof [y]_~ = > [x]_~ Subsetequalto [y]_~ --- (3) \r\n similarly, let z elementof [y]_~ = > z ~ y --- (4) Also by hypothesis x ~ y i.e, y ~ x --- (5) Now from (4), (5) , we have z ~ x = > z elementof [x]_~ (By transitivity of ~) So, we observed that z elementof [y]_~ = > z elementof [x]_~ = > [y]_~ Subsetequalto [x]_~ --- (6) Now from (3) and (6), we have