Example 2

Let $\mathrm{E}$, $\mathrm{F}$ be two sets and let $\mathrm{R}$ be the equivalence relation on $\mathrm{E} \times \mathrm{F}$ associated with the mapping $\mathrm{pr}_1$ of $\mathrm{E} \times \mathrm{F}$ onto $\mathrm{E}$. The equivalence classes with respect to $\mathrm{R}$ are the sets of the form $\{x\} \times \mathrm{F}$, where $x \in \mathrm{E}$; the mapping $x \to \{x\} \times \mathrm{F}$ is a bijection of $\mathrm{E}$ onto $(\mathrm{E} \times \mathrm{F})/\mathrm{R}$.

Let $\mathrm{R}$ be an equivalence relation on a set $\mathrm{E}$. The quotient set $\mathrm{E}/\mathrm{R}$ is a subset of $\mathfrak{P}(\mathrm{E})$, and the identity mapping of $\mathrm{E}/\mathrm{R}$ is a partition of $\mathrm{E}$ (§ 4, no. 7); for if $\mathrm{G}$ is the graph of $\mathrm{R}$, we have $x \in \mathrm{G}(x)$ for all $x \in \mathrm{E}$, and if two equivalence classes $\mathrm{G}(x)$ and $\mathrm{G}(y)$ have a common element $z$, then $\mathrm{R}\{x,\ z\}$ and $\mathrm{R}\{y,\ z\}$, so that $\mathrm{G}(x) = \mathrm{G}(y)$. Furthermore, the relation

is equivalent to $\mathrm{R}\{x,\ y\}$.

¶ Conversely, let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a partition of a set E. It is immediately verified that the relation $(\exists \iota)(\iota \in \mathrm{I}$ and $x \in \mathrm{X}_\iota$ and $y \in \mathrm{Y}_\iota)$ is an equivalence relation R on E; the equivalence classes with respect to R are just the sets $\mathrm{X}_\iota$ of the partition, and the mapping $\iota \to \mathrm{X}_\iota$ is a bijection of I onto E/R. Every subset S of E such that, for each $\iota \in \mathrm{I}$, the set $\mathrm{S} \cap \mathrm{X}_\iota$ consists of a single element is called a system of representatives (or a transversal) of the equivalence classes with respect to R. This name is also used to denote any injection of a set K into E such that the image of K under this injection is a system of representatives of the equivalence classes with respect to R; for example, any section of E with respect to R.