In textbooks, I've always seen the notion of quotient set defined with equivalence relations, that is: if $R$ is an equivalence relation on a set $X$, we can define the quotient set $X/R = \{[x]_R \mid x \in X\}$ where $[x]_R = \{y \in X \mid (x,y) \in R\}$ is the equivalence class represented by $x$.
However, isn't the definition of $X/R$ also valid even if $R$ is not an equivalence relation on the set $X$? For instance, could it be possible to define $\Bbb{R}\,/\!≤$ $\;= \{[x]_{≤} \mid x \in \Bbb R\}$, where $[x]_{≤} = ]-\infty, x]$ (and $≤$ the usual order on the real numbers)?
Thank you!
What a nice idea. For any relation $R$ on a set $X$, you might more generally define
$$X / R \equiv \{ \{ y \in X : y R x\} : x \in R \}$$
(In order theory, these are related to the notion of ideals.)
This construction has the property that if $R$ is a partial order on $X$, then $X/R \cong X$, and if $R$ is an equivalence relation then $X/R$ is the optimal way of dividing $X$ into groups such that (1) each partition contains members that are all $R$-related to each other, and (2) each $x\in X$ appears in exactly one group.