# What are isomorphic partitions?

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.

• Every equivalence relation determines a partition (and vice versa). You have to use the equivalence relation to define the partition, and use the same idea to define what it means for partitions to be isomorphic. – user458276 Dec 14 '18 at 6:21
• Yes, but what does it mean for partitions to be isomorphic? – childishsadbino Dec 14 '18 at 6:24
• Also, when they say $(y,z)$ is in $R_1$, does it mean they are in the same equivalence class, which in turn implies that $f(x), f(y)$ are in the same equivalence class in $R_2$? So, assuming that to be true, for partitions to be isomorphic, they would take elements from a fixed subset of $P_1$ only to a fixed unique set of $P_2$, right? – childishsadbino Dec 14 '18 at 6:30
• Wait, is it that the number of subsets in partitions are equal? – childishsadbino Dec 14 '18 at 6:53
• Well, finding the definition is exactly what point (a) is about; and it even gives you a way to check your answer by doing point (b). – Arnaud D. Dec 14 '18 at 16:07