Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) is

Given set $A = \{1, 2, 3\}$ with three elements.

Need to find: the number of equivalence relations that must include the pair $(1, 3)$.

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Any equivalence relation must satisfy three properties:

Reflexive: Every element relates to itself, so must have $(1,1)$, $(2,2)$, $(3,3)$

Symmetric: If $a$ relates to $b$, then $b$ relates to $a$

Transitive: If $a$ relates to $b$ and $b$ relates to $c$, then $a$ relates to $c$

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Since the relation must contain $(1,3)$:

From the given condition: $(1,3)$

From symmetry: $(3,1)$

From reflexivity: $(1,1)$, $(2,2)$, $(3,3)$

The relation so far: $\{(1,1), (2,2), (3,3), (1,3), (3,1)\}$

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Since $1$ and $3$ are equivalent, they form a group. Element $2$ has two choices:

Case 1: Element $2$ stays separate

Partition: $\{1, 3\}$, $\{2\}$

Relation: $\{(1,1), (2,2), (3,3), (1,3), (3,1)\}$

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Case 2: Element $2$ joins the group with $1$ and $3$

Partition: $\{1, 2, 3\}$

Need to add: $(1,2)$, $(2,1)$, $(2,3)$, $(3,2)$

Relation: $\{(1,1), (2,2), (3,3), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)\}$

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Once $(1,3)$ is in the relation, elements $1$ and $3$ are locked together. Element $2$ can either join them or stay separate.

Total number of equivalence relations containing $(1,3)$ = $2$