Let A = {1, 2, 3}. The number of equivalence | vedclass

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(C) For a relation $R$ on $A = \{1, 2, 3\}$ to be an equivalence relation containing $(1, 2)$,it must satisfy reflexivity,symmetry,and transitivity.Si

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$2$

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(C) For a relation $R$ on $A = \{1, 2, 3\}$ to be an equivalence relation containing $(1, 2)$,it must satisfy reflexivity,symmetry,and transitivity.Since it is reflexive,$(1, 1), (2, 2), (3, 3) \in R$.Since it contains $(1, 2)$ and is symmetric,$(2, 1) \in R$.Now we have $R_0 = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$. This is an equivalence relation.To add more elements while maintaining equivalence,we must consider the partitions of $A$. The partition corresponding to $R_0$ is $\{\{1, 2\}, \{3\}\}$.The only other partition of $A$ that contains the subset $\{1, 2\}$ is the trivial partition $\{\{1, 2, 3\}\}$.The equivalence relation corresponding to the partition $\{\{1, 2, 3\}\}$ is the universal relation $A 	imes A$,which contains $(1, 2)$.Thus,there are $2$ such equivalence relations: $\{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ and $A 	imes A$.

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

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