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

$\begin{array}{1 1} 1 \\ 2 \\ 3 \\ 4 \end{array}$

Toolbox:
• A relation R in A is an equivalence relation if it is,reflexive,symmetric and transitive.
The smallest equivalence relation containing (1,2) is $R=\{(1,1),(2,2),(3,3),(1,2)(2,1)\}$
To get another relation which is equivalence can be done by adding the remaining pairs $(2,3),(3,2),(1,3),(3,1)$ to get the universal relation.
This shows that there are only 2 equivalence relation containing (1,2)
edited Mar 20, 2013