Let A = \ 1, 2, 3, 4\ . The number of equivalence relations on the...

The number of equivalence relations on a set of $n$ elements is given by the $n$-th Bell number $B_n$.

For $n=4$, the total number of equivalence relations is $B_4 = 15$.

Let us find the number of equivalence relations that contain the ordered pair $(1,2)$. Since an equivalence relation is symmetric and transitive, containing $(1,2)$ implies that $1$ and $2$ must belong to the same equivalence class.

We can treat $\{1,2\}$ as a single element. The problem then reduces to finding the number of equivalence relations on a set of $3$ elements (namely, the block $\{1,2\}$, $3$, and $4$).

The number of equivalence relations on a $3$-element set is $B_3 = 5$.

Therefore, the number of equivalence relations on $A$ that do not contain $(1,2)$ is the total number of equivalence relations minus the number of equivalence relations containing $(1,2)$.

Required number $= 15 - 5 = 10$.

Answer: $10$