Let f A rightarrow B be a surjective map of sets Prove that

Let f: A rightarrow B be a surjective map of sets. Prove that the relation a ~ b if and only if f (a) = f (b) is an equivalence relation and find its equivalence classes.

Solution

Let f: A to B be a surjective map of sets.

In other words if b is in B, then there is an \"a\" in A such that f(a) = b, for all b

Define the relation ~ as a~b if f(a) =f(b)

To prove that ~ is an equivalence relation.

~ is reflexive as f(a) = f(a) for all a

~ is symmetric as if f(a) = f(b) then f(b) = f(a). Hence a~b implies b~a

Let a~b and b~c be true.

Then f(a) = f(b) and f(b) = f(c)

So f(a) = f(c) and a~c holds good

Thus transitive also

So the relation is an equivalence relation