The "characterizing property" for ordered pairs is that two ordered pairs are equal if and only if their respective components are equal. We use a set-theoretic implementation of ordered pairs, where ordered pairs are defined as certain sets. The most common implementation is the Kuratowski ordered pair, but there are others. Also, the characterizing property for Cartesian products can be defined in terms of category theory. Again, there is a set-theoretic implementation of that too. That raises the question, what is the characterizing property for functions? That is, what is the property that an object $f$ has to have in order for it to be a function from domain $A$ to codomain $B$? Also, bonus question, is there an implementation of the function idea such that its graph is not a set of ordered pairs, for any ordered pair function $OP$? I once asked if there is an implementation of Cartesian products of sets which is not a set of ordered pairs, and it turns out there is such an implementation.
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A function is formally given by its graph. Thus, two functions are equal if, and only if, they have the same graph.
Alek Fröhlich
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