Subsection 4.3.4 (0034): Ordered Pairs

Table of Contents
Part 2: Sets
Chapter 4: Constructions With Sets
Section 4.3: Operations With Sets
Subsection 4.3.4: Ordered Pairs (cite)

4.3.4 Ordered Pairs

Let $A$ and $B$ be sets.

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

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

Proposition 4.3.4.1.2 ▶ Properties of Ordered Pairs

Let $A$ and $B$ be sets.

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