Question

Let `S` be a relation on the set `R` of all real numbers defined by `S={(a , b)RxR: a^2+b^2=1}dot` Prove that `S` is not an equivalence relation on `Rdot`

Answer 1

`S = {(a,b): a,b in R and a^2+b^2 =1}`
`a^2+a^2 = 1 =>2a^2 = 1`.
There can be values of `a in R` such that `2a^2 != 1`
`:. (a,a) != S`.
`:. S` is not reflexive.

For a relation to be equivalence, it should be reflexive, symmetric and transitive.
But, as `S` is not reflexive, it can not be an equivalence relation.