The number of equivalence relation on the set $\{1, 2, 3\}$ containing $(1, 2)$ and $(2, 1)$ is

An equivalence relation on the set $\{1, 2, 3\}$ must satisfy three properties:

• Reflexive: $(1,1)$, $(2,2)$, $(3,3)$ must be included
• Symmetric: If $(a,b)$ is included, then $(b,a)$ must be included
• Transitive: If $(a,b)$ and $(b,c)$ are included, then $(a,c)$ must be included

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The relation must contain $(1,2)$ and $(2,1)$.

By the reflexive property, the relation must also contain $(1,1)$, $(2,2)$, and $(3,3)$.

Therefore, the mandatory pairs are: $(1,1)$, $(2,2)$, $(3,3)$, $(1,2)$, $(2,1)$

This means elements $1$ and $2$ are in the same equivalence class.

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For element $3$, there are two possibilities:

Case 1: Element $3$ forms its own equivalence class.

The equivalence classes are $\{1, 2\}$ and $\{3\}$.

The relation is $\{(1,1), (2,2), (3,3), (1,2), (2,1)\}$

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Case 2: Element $3$ is in the same equivalence class as $1$ and $2$.

The equivalence class is $\{1, 2, 3\}$.

Since $3$ must be related to both $1$ and $2$, the relation must contain $(1,3)$, $(3,1)$, $(2,3)$, and $(3,2)$ by symmetry and transitivity.

The relation is $\{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)\}$

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There are exactly $2$ equivalence relations on the set $\{1, 2, 3\}$ containing $(1,2)$ and $(2,1)$.