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ordered pair (Definition)

For any sets $ a$ and $ b$, the ordered pair $ (a,b)$ is the set $ \{\{a\}, \{a,b\}\}$.

The characterizing property of an ordered pair is:

$\displaystyle (a,b) = (c,d) \iff a=b$    and  $\displaystyle c=d, $
and the above construction of ordered pair, as weird as it seems, is actually the simplest possible formulation which achieves this property.



"ordered pair" is owned by djao.
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ordered tuplet (Definition) by rspuzio
$(a,b)=(c,d)$ if and only if $a=c$ and $b=d$ (Proof) by Wkbj79
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Cross-references: property
There are 43 references to this entry.

This is version 4 of ordered pair, born on 2001-10-19, modified 2002-03-04.
Object id is 358, canonical name is OrderedPair.
Accessed 25964 times total.

Classification:
AMS MSC: 03-00 (Mathematical logic and foundations :: General reference works )
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Discussion
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proof by nctu on 2004-04-03 23:24:30
(a,b)=(c,d) <==> a=c, b=d
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explaination by akrowne on 2001-10-28 02:00:44
For all those out there who don't understand why an ordered pair must be defined like this, try exchanging a and b in the definition above, and you'll see that (a,b) != (b,a).

-apk
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