While writing a proof for below statement, I stuck at representing equivalence relation as a function.
Let $f : A → B $ be a function and let G be an equivalence relation in B.
Prove that the preimage of G i.e $f^\vee(G) = f^{-1}\circ G \circ f$.
I am thinking of representing equivalence relation as an ordered pair $G \times G$. But it is not a function.
Lets take B = { 1, 2, 4 } and let R be the relation representing even numbers. Now I will represent R as {(2,2), (4,4), (2,4), (4,2)}. R is not a function since in the order pairs (2,2) and (2,4), 2 is mapped to both 2 and 4.
In the above case how can I represent R as a function.