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2. Universal set - Wikipedia

   en.wikipedia.org/wiki/Universal_set

   There are set theories known to be consistent (if the usual set theory is consistent) in which the universal set V does exist (and is true). In these theories, Zermelo's axiom of comprehension does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.

3. Set (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Set_(mathematics)

   A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...

4. List of set identities and relations - Wikipedia

   en.wikipedia.org/wiki/List_of_set_identities_and...

   In constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).

5. Euler diagram - Wikipedia

   en.wikipedia.org/wiki/Euler_diagram

   For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves.

6. Intersection (set theory) - Wikipedia

   en.wikipedia.org/wiki/Intersection_(set_theory)

   When is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist.

7. Initial and terminal objects - Wikipedia

   en.wikipedia.org/wiki/Initial_and_terminal_objects

   Dually, a universal morphism from U to X is a terminal object in (U ↓ X). The limit of a diagram F is a terminal object in Cone(F), the category of cones to F. Dually, a colimit of F is an initial object in the category of cones from F. A representation of a functor F to Set is an initial object in the category of elements of F.