Let `f:X->Y` be a function. Define a relation `R` in `X` given by `R={(a,b):f(a)=f(b)}.` Examine whether `R` is an equivalence relation or not.
`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.
`f:X->Y`
`R={(a,b):f(a)=f(b)}`
a)Reflexive
`r(a,a) forall a in X`
`f(a)=f(a)`
R is reflexive.
b)Symmetric
`(a,b)in R`
then`(b,a)inR`
`f(a)=f(b)`
`f(b)=f(a)`
`(b,a)inR`
R is symmatric
c)Transitive
`(a,b),(b,c)inR`
then`(a,c)inR`
`f(a)=f(b)`
`f(b)=f(c)`
`f(a)=f(c)`
`R` is an equivalence relation.