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Find the number of equivalence relations on $\{1,2,3,4\}$ that contains $\{ (1,2), (3,4)\}$.

What I've been doing:

I tried to find the matrix for that relation:

$\begin{pmatrix} 1 & 1 & * & *\\ 1 & 1 & * & * \\ * & * & 1 & 1 \\ * & * & 1 & 1 \end{pmatrix}$

Let $\Delta$ be the main diagonal for that matrix, so the number of ways to put $1$ or $0$s on $\Delta$ is $1^4$ (only one, because it's reflexive).

Then the number of ways to put $1$ or $0$s on $\bar \Delta$ (the numbers outside the main diagonal), such that the relation is simetric is $2^4$ ($4$ = the number of stars in my matrix such that they're eiher $0$ or $1$).

Now I don't have a clue how to count the number of transitive relations. My idea was to count all the reflexive, simetric and transitive relations and then multiply them (because of the product rule).

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There is a one-to-one correspondence between equivalence relations on a set and partitions of that set. Since you already know $1$ and $2$ are in the same part and $3$ and $4$ are as well, you are left with just two possible partitions: the trivial partition and $\{\{1,2\},\{3,4\}\}$.

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