Question

Write the smallest equivalence relation on the set $\{1,2,3\}$.

Answer 1

Determine the smallest equivalence relation on the set $\{1,2,3\}$.

An equivalency relation is one that is reflexive, symmetric, and transitive:

$$\begin{gathered} A=\{1,2,3\} \\ \Rightarrow A\times A=\{1,2,3\}\times \{1,2,3\} \\ \therefore A\times A=\left\{(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)(3,1)(3,2)(3,3)\right\} \end{gathered}$$

Consider $R=\left\{\right((1,1),(2,2),(3,3)\}$ is the smallest equivalence relation on the set $A=\{1,2,3\}$.

Since $x\in A,(x,x)\in R\to$ $R$ is Reflexive.

Also relation $R$ is symmetric defined as $\left(x,y\right)\in R\Rightarrow \left(y,x\right)\in R$ for all $x,y,z\in A$.

If $(x,y)\in R,(y,z)\in R\Rightarrow (x,z)\in R$ for all $x,y,z\in A$. the relation $R$ is transitive.

Hence, the relation $R$ is the smallest equivalence relation on given set.