For x,y ∈ R def i ne x ∼ y to mean that |x| = |y|. Prove that ∼ is an equivalence relation in R. Describe [0],[5], and [−5].
Answer (1)
13th Nov, 2020
Hello Siddhant!
a) We know that, |x|= |x| => x~x for all x∈R
So, ~ is reflexive relation on R.
b) We have, |y|= |x|=> y~x
i.e, if x~y then y~x, so, ~ is symmetric on R.
c) Let x~y and y~z be arbitrary relations on R
=> |x|=|y| and |y|=|z|
=>|x|=|z|=> x~z
So, ~ is transitive on R as when x~y and y~z, we get x~z as a true statement.
Hence, it is proved that ~ is an equivalence relation on R.
Hope you understood:)
Comments (0)
