Kuratowski ordered pairs

See also: Mathematical Analysis — index

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An unordered pair is simply a 2-element set, say  [;\{a\ \ b\};],  where  [;a \ne b ;]  (note that when  [; a = b ;]  then  [;\{a\ \ b\};],  is a 1-element set).

Notation for ordered pairs is  [;(a\ \ b);],  where  [;a\ \ b;]  are arbitrary--they can be different or they may coincide. The ordered pairs satisfy the following axiom:

[; :\quad\quad (a\ \ b)\ =\ (c\ \ d)\ \ \ \Leftrightarrow\ \ \ (a=c\ \ \&\ \ b=d) ;]

An elegant constructive definition of an ordered pair in the set-theoretical term was given by Kazimierz Kuratowski:

[; :\quad\quad (a\ \ b)\ :=\ \{ \{ a \}\ \{a\ \ b\}\} ;]

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Now we can define an ordered triple [;(a\ \ b\ \ c);]. It can be done in many different ways. For many purposes the details of an exact constructive definition are irrelevant. Then it is important only that ordered triples satisfy the following axiom:

[; :\quad\quad (a\ \ b\ \ c)\ =\ (x\ \ y\ \z)\ \ \ \Leftrightarrow\ \ \ (a=x\ \ \&\ \ b=y\ \ \&\ \ c=z) ;]

A definition in the Kuratowski's style is:

[; :\quad\quad (a\ \ b\ \ c)\ :=\ \{ \{ a \}\ \{a\ \ b\}\ \{a\ \ b\ \ c\}\} ;]

We may use Kuratowski pairs to define an ordered triple as a sequence of length 3:

[; :\quad\quad (a\ \ b\ \ c)\ :=\ \{ (0\ a)\ \ (1\ b)\ \ (2\ c) \} ;]

or as a string (again of length 3):

[; :\quad\quad (a\ \ b\ \ c)\ := (a\ \ (b\ \ c)) ;]

where  [;a;]  is called the head of the string, and [;(b\ \ c);] is the tail of the string.

Different types of the triples of the same three elements [;a\ b\ c;] are not equal. Thus be consistent, don't mix them. And if you do then use different notation for each type.