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2. Equivalence relation - Wikipedia

   en.wikipedia.org/wiki/Equivalence_relation

   In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number is equal to itself (reflexive).

3. Equality (mathematics) - Wikipedia

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

   Ernst Zermelo, a contributer to modern Set theory, was the first to explicitly formalize set equality in his Zermelo set theory (now obsolete), by his Axiom der Bestimmtheit. [ 31 ] Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.

4. Equivalence class - Wikipedia

   en.wikipedia.org/wiki/Equivalence_class

   In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements a {\displaystyle a} and b {\displaystyle b} belong to the same equivalence class if, and only if , they are ...

5. 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 ).

6. Equinumerosity - Wikipedia

   en.wikipedia.org/wiki/Equinumerosity

   In some other systems of axiomatic set theory, for example in Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, relations are extended to classes. A set A is said to have cardinality smaller than or equal to the cardinality of a set B, if there exists a one-to-one function (an injection) from A into B.

7. Algebra of sets - Wikipedia

   en.wikipedia.org/wiki/Algebra_of_sets

   The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".

8. Equivalence (measure theory) - Wikipedia

   en.wikipedia.org/wiki/Equivalence_(measure_theory)

   So the counting measure has only one null set, which is the empty set. That is, N μ = { ∅ } . {\displaystyle {\mathcal {N}}_{\mu }=\{\varnothing \}.} So by the second definition, any other measure ν {\displaystyle \nu } is equivalent to the counting measure if and only if it also has just the empty set as the only ν {\displaystyle \nu ...

9. List of types of sets - Wikipedia

   en.wikipedia.org/wiki/List_of_types_of_sets

   8 Ways of defining sets/Relation to descriptive set theory. 9 More general objects still called sets. 10 See also. Toggle the table of contents. List of types of sets.