Example 2

Let $\mathrm{R}\}x,\ y\{$ be an order relation between $x$ and $y$, and let E be a set such that $x \in \mathrm{E}$ implies $\mathrm{R}\}x,\ x\{$ (notice that the empty set satisfies this condition). The relation "$\mathrm{R}\}x,\ y\{$ and $x \in \mathrm{E}$ and $y \in \mathrm{E}$" is then an order relation on E, as is immediately verified; it is called the order relation induced by $\mathrm{R}\}x,y\{$ on E (cf. no. 4). By abuse of language, the phrase "the relation $\mathrm{S}\}x,\ y\{$ is an order relation between elements of E" is often used in place of "the relation ($\mathrm{S}\}x,\ y\{$ and $x \in \mathrm{E}$ and $y \in \mathrm{E}$) is an order relation on E". For example, if A is a set, the relation "$\mathrm{X} \subset \mathrm{Y}$ and $\mathrm{X} \subset \mathrm{A}$ and $\mathrm{Y} \subset \mathrm{A}$" is an order relation between subsets of A.