Let $R_1, R_2 ∈ R(X)$ be equivalence relations on $X$.
Define $R_1$ and $R_2$ to be isomorphic if there exists a bijection $f : X → X$ such that the following holds: For all $y, z ∈ X : (y, z) ∈ R_1$ if and only if $(f(y), f(z)) ∈ R_2$
(a) Define what is means for two partitions $P_1, P_2 ∈ P(X)$ to be isomorphic. (An answer to this is correct if it lets you prove the next part.)
(b) Prove two equivalence relations $R_1$ and $R_2$ are isomorphic if and only if the partitions $φ(R_1)$ and $φ(R_2)$ are isomorphic. (Here φ is the bijection from the previous problem.)
(c) Let X = {1, 2, 3, 4, 5}. Up to isomorphism, how many equivalence relations are there on X?
My main issue is that I do not understand what an isomorphic partition is. Only when I understand this can I begin to even answer part (b) or (c).
NOTE: I do not get why this post was downvoted when all I asked for is direction to be able to solve this question and even commented on what I understand so far.
When P is a partition, the partition part containing
a, a/P = { x : exists A in P with x,a in A }.
Two partitions P,Q are isomorphic when
exists bijection f:X -> X with for all x,y in X,
x/P = y/P iff f(x)/Q = f(y)/Q.