Question

Let `f : X->Y`be a function. Define a relation R in X given by `R = {(a , b): f(a) = f(b)}`. Examine if R is an equivalence relation.

Answer 1

`R = {(a,b): f(a) = f(b)}`
As `f(a) = f(a)`
`:. (a,a) in R`
`:. R` is reflexive.
If `f(a) = f(b)`
Then, `f(b) = f(a)`
Thus, `(b,a) in R`
So, if `(a,b) in R`, then `(b,a) in R`
`:. R` is ymmetric.
If `(a,b) in R`,
Then, `f(a) = f(b)`->(1)
If `(b,c) in R`,
Then, `f(b) = f(c)`->(2)
From (1) and (2),
`f(a) = f(c)`
`:. (a,c) in R`
`:. R` is transitive.
As `R` is reflexive, summetric and transitive, `R` is an equivalence relation.