21. Given a partition $\{X_{\alpha}\}_{\alpha\in I}$ of $X$, show that the relation $R\subseteq X\times X$ defined by $((a,b)\in R \Leftrightarrow \exists \alpha\in I \text{ such that both } a,b\in X_{\alpha})$ is an equivalence relation on $X$ whose equivalence classes are precisely the sets $X_{\alpha}$ of the partition.

Proof: Let $X$ be a nonempty set. Then, by deifnition, $R\subseteq X\times X$ is an equivalence relation on $X$ if and only if for all a ∈ $X$, (a, a) ∈ R, if, for any a, b ∈ $X$, (a, b) ∈ R, then (b, a) ∈ R, and if, for any a, b, c ∈ $X$, (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

1. Reflexivity: (m, n)~(m, n)

2. Symmetry: (m, n)~(p,q) → (p, q)~(m, n)

3. Transitivity: (m, n)~(p, q)~(r, s), show (m, n)~(r, s)

*know what I have to do but not sure of the equation we are using