Set theory MOC

Subset

Let be sets. A subset is a set whose elements are all elements of , set i.e.

A proper subset is is a subset that is not equal to its superset, i.e.

Universal property

Adopting a structuralist perspective, let denote the membership predicate so that . A subset along with its natural inclusion is characterized up to unique bijection by the following universal property:

. If is a set and is a function such that , then there exists a unique function such that , i.e.

Proof

Clearly by construction. If , then so we can take .

This may be rephrased as a fibre product for a more general Subobject via a Subobject classifier, generalizing this construction to an arbitrary Elementary topos as well as some other categories.

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