Different Representatives for Equivalence Classes

Recall from the Equivalence Classes page that if $X$ is a set and $\sim$ is an equivalence relation on $X$ then if $x \in X$, the equivalence class of $x$ is defined to be:

(1)

\begin{align} \quad [x] = \{ y \in X : x \sim y \} \end{align}

Suppose that $x_1, x_2 \in X$ and $x_1 \sim x_2$. We will show that then $[x_1] = [x_2]$.

Theorem 1: Let $X$ be a set and let $\sim$ be an equivalence relation on $X$. If $x_1, x_2 \in X$ and $x_1 \sim x_2$ then $[x_1] = [x_2]$.

• Proof: Let $x \in [x_1]$. Then $x \sim x_1$. Since $x_1 \sim x_2$ and $\sim$ is transitive we have that $x \sim x_2$. So $x \in [x_2]$. Hence:

(2)

\begin{align} \quad [x_1] \subseteq [x_2] \quad (*) \end{align}

• Now let $x \in [x_2]$. Then $x \sim x_2$. Since $x_1 \sim x_2$, by symmetry we have that $x_2 \sim x_1$. So by transitivity we have that $x \sim x_1$. So $x \in [x_1]$. Hence:

(3)

\begin{align} \quad [x_2] \subseteq [x_1] \quad (**) \end{align}

• From $(*)$ and $(**)$ we conclude that $[x_1] = [x_2]$. $\blacksquare$