Definition:Quotient Mapping

Definition

Let $\mathcal R \subseteq S \times S$ be an equivalence on a set $S$.

Let $\left[\!\left[{s}\right]\!\right]_\mathcal R$ be the $\mathcal R$-equivalence class of $s$.

Let $S / \mathcal R$ be the quotient set of $S$ determined by $\mathcal R$.

Then $q_\mathcal R: S \to S / \mathcal R$ is the quotient mapping induced by $\mathcal R$, and is defined as:

$q_\mathcal R: S \to S / \mathcal R: q_\mathcal R \left({s}\right) = \left[\!\left[{s}\right]\!\right]_\mathcal R$

Effectively, we are defining a mapping on $S$ by assigning each element $s \in S$ to its equivalence class $\left[\!\left[{s}\right]\!\right]_\mathcal R$.

If the equivalence $\mathcal R$ is understood, $q_\mathcal R \left({s}\right)$ can be written $q \left({s}\right)$.

Also known as

The quotient mapping is often referred to as:

the canonical surjection from $S$ to $S / \mathcal R$

the natural surjection from $S$ to $S / \mathcal R$

the classifying map or classifying mapping (as it classifies the elements of $S$ into those various equivalence classes)

Some sources denote the quotient mapping by $\natural_\mathcal R$. This is logical, as $\natural$ is the "natural" sign in music.

Also see

• Quotient Mapping is Surjection
• Induced Equivalence

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 3$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 8$: Functions
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 10$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.4$: Example $2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 17$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 6$