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:)