Quotient set

Let $\sim$ be an equivalence relation on $𝐴$ , and for each $𝑎 \in 𝐴$ let $[𝑎]$ denote its equivalence class. Then the quotient set $𝐴 / \sim$ is the set of all such equivalence classes naïve

𝐴 / \sim = {[𝑎] : 𝑎 \in 𝐴}

with the natural projection

𝜋 : 𝐴 ↠ 𝐴 / \sim 𝑎 \mapsto [𝑎]

Universal property

The quotient set with canonical projection $(𝐴 / \sim, 𝜋)$ is characterized up to unique isomorphism by the universal property:

$𝑎 \sim 𝑏 ⟹ 𝜋 (𝑎) = 𝜋 (𝑏)$ . If $𝐵$ is a set a set and $𝑓 : 𝐴 \to 𝐵$ is a function with $𝑎 \sim 𝑏 ⟹ 𝑓 (𝑎) = 𝑓 (𝑏)$ , then there exists a unique function $¯ 𝑓 : 𝐴 / \sim \to 𝐵$ so that $𝑓 = ¯ 𝑓 𝜋$ , i.e.

Proof

This is fairly trivial to prove.

This notion is treated more generally by the Quotient object.

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Table of Contents

Quotient set

Universal property

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