Quotient by an equivalence relation

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Definition

Let $X$ be a set and let $\sim\subseteq X\times X$ be an equivalence relation on $X$. Then the quotient of $X$ by $\sim$, denoted $X/\sim$ or $\frac{X}{\sim}$ is defined as follows:

• $\frac{X}{\sim}:\eq\{[x]\ \vert\ x\in X\}$ where $[x]$ denotes the equivalence class of $x$.

Yes, $\frac{X}{\sim}$ is the set of equivalence classes, it is that simple.

Canonical projection

With $X$, $\sim$ and $\frac{X}{\sim}$ we also get a map:

• $\pi:X\rightarrow\frac{X}{\sim}$ given by $\pi:x\mapsto [x]$
  • Other commonly used letters include: $p$ and $\rho$

Claim 1: this map is a surjection

Proof of claims

Claim 1: $\pi:X\rightarrow\frac{X}{\sim}$ is a surjection

We wish to show: $\forall y\in\frac{X}{\sim}\exists x\in X[\pi(x)\eq y]$

• Let $y\in\frac{X}{\sim}$ be given
  • Choose $a\in y$ (so now we may write $[a]\eq y$. Any $a$ will do.
    • Note that the equivalence classes are either equal or disjoint, suppose $b\in y$, this means $a\sim b$, so $[a]\eq [b]$
  • Choose $x:\eq a$
    • Then $\pi(x)\eq [x]:\eq[a]\eq y$
• Since $y$ was arbitrary we have shown this for all $y\in\frac{X}{\sim}$

See also

• Axiom of choice