Let
$A=\{1,2,3,4,5,6,7,8\}$.
$S=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8)\}$.
$T=\{(1,2),(2,1),(5,4),(4,5),(6,2),(6,5))\}$.
How many equivalence relations R on A exist such that R$\subseteq$S$\cup$T?
I got 5, but the answer is 4.
These are the partitions of the set I got:
$R1=\{\{1,2,6,5,4\},\{3\},\{7\},\{8\}\}$
$R2=\{\{1,2,6,5,4\},\{3,7\},\{8\}\}$
$R3=\{\{1,2,6,5,4\},\{3\},\{7,8\}\}$
$R4=\{\{1,2,6,5,4\},\{3,7,8\}\}$
$R5=\{\{1,2,6,5,4\},\{3,8\},\{7\}\}$
What did I do wrong?
None of the partitions you list corresponds to an equivalence relation that contains $S\cup T$, since neither $(4,6)$ nor $(6,4)$ is in $S\cup T$.
We can disregard $S$, since an equivalence relation is symmetric and must include all elements of $S$ anyway. From $T$ we can only use pairs whose transposition is also in $T$, because the equivalence relation needs to be symmetric. Thus we want to know which equivalence relations use only the pairs $(1,2),(2,1),(5,4),(4,5)$ (in addition to those in $S$). We can independently choose whether to include $(1,2)$, $(2,1)$ and whether to include $(5,4)$, $(4,5)$, so that’s $2^2=4$ possible choices.