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ordered pair

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

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