Subsection 2.3.4 (0034): Ordered Pairs

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.3: Operations With Sets
Subsection 2.3.4: Ordered Pairs (cite)

2.3.4 Ordered Pairs

Let $A$ and $B$ be sets.

Definition 2.3.4.1.1 ▶ Ordered Pairs

The ordered pair associated to $A$ and $B$ is the set $\webleft (A,B\webright )$ defined by

\[ \webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft\{ A\webright\} ,\webleft\{ A,B\webright\} \webright\} . \]

Proposition 2.3.4.1.2 ▶ Properties of Ordered Pairs

Let $A$ and $B$ be sets.

Uniqueness. Let $A$, $B$, $C$, and $D$ be sets. The following conditions are equivalent:
1. We have $\webleft (A,B\webright )=\webleft (C,D\webright )$.
2. We have $A=C$ and $B=D$.

Proof of Proposition 2.3.4.1.2.

Item 1: Uniqueness

See Theorem 1.2.3 of [Ciesielski, Set Theory for the Working Mathematician].

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