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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. Set theory - Wikipedia

   en.wikipedia.org/wiki/Set_theory

   Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole.

6. Simple theorems in the algebra of sets - Wikipedia

   en.wikipedia.org/wiki/Simple_theorems_in_the...

   The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.

7. 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.

8. Naive set theory - Wikipedia

   en.wikipedia.org/wiki/Naive_set_theory

   For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset A of U, the complement of A (in U) is defined as

9. Universe (mathematics) - Wikipedia

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

   In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation. In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem.