Equivalence of Definitions of Ordered Pair

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Theorem

The following definitions of an ordered pair are equivalent:

Informal Definition

An ordered pair is a two-element set together with an ordering.

In other words, one of the elements is distinguished above the other - it comes first.

Such a structure is written:

$\tuple {a, b}$

and it means:

first $a$, then $b$.

Kuratowski Formalization

The concept of an ordered pair can be formalized by the definition:

$\tuple {a, b} = \set {\set a, \set {a, b} }$

This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.


Proof

Equality of Ordered Pairs

From Equality of Ordered Pairs, we have that:

$\set {\set a, \set {a, b} } = \set {\set c, \set {c, d} } \iff a = c, b = d$

hence showing that the Kuratowski formalization fulfils the requirement for equality.


Existence of Cartesian Product

Let $A$ and $B$ be sets.


Suppose that either $A$ or $B$ is empty.

Then:

\(\displaystyle A \times B\) \(=\) \(\displaystyle \set {\tuple {a, b} : a \in A \land b \in B}\) Definition of Cartesian Product
\(\displaystyle \) \(=\) \(\displaystyle \O\)

By the Axiom of the Empty Set $A \times B$ is a set.


Now suppose that $A$ and $B$ are both non-empty.


Let $a \in A$ and $b \in B$.

By Singleton of Element is Subset:

$\set a \subseteq A$ and $\set b \subseteq B$

Therefore by Set Union Preserves Subsets:

$\set {a, b} \subseteq A \cup B$

$A \cup B$ is a set by Union of Small Classes is Small.


Because $\set a \subseteq A \subseteq A \cup B$, it follows that:

$\set a, \set {a, b} \in \powerset {A \cup B}$


where $\powerset {A \cup B}$ is the power set of $A \cup B$.

This a set by the Axiom of Powers.


Thus:

$\set {\set a, \set {a, b} } \subseteq \powerset {A \cup B}$


and so by definition of the power set:

$\set {\set a, \set {a, b} } \in \powerset {\powerset {A \cup B} }$


It has been shown that:

$\set {\tuple {a, b}: a \in A \land b \in B} \subseteq \powerset {\powerset {A \cup B} }$


Applying the axiom of specification and the axiom of extension, the unique set $A \times B$ is created which consists exactly of ordered pairs $\tuple {a, b}$ such that $a \in A$ and $b \in B$.

Thus it has been demonstrated that if $A$ and $B$ are non-empty, then the cartesian product $A \times B$ exists and is non-empty.


Finally, in Subset of Cartesian Product it is demonstrated that every set of ordered pairs is a subset of the cartesian product of two sets.

$\blacksquare$


Sources