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2. Equinumerosity - Wikipedia

   en.wikipedia.org/wiki/Equinumerosity

   Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...

3. Equality (mathematics) - Wikipedia

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

   In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol ≅ {\displaystyle \cong } ) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects.

4. Equivalence class - Wikipedia

   en.wikipedia.org/wiki/Equivalence_class

   The set of all equivalence classes in with respect to an equivalence relation is denoted as /, and is called modulo (or the quotient set of by ). [3] The surjective map x ↦ [ x ] {\displaystyle x\mapsto [x]} from X {\displaystyle X} onto X / R , {\displaystyle X/R,} which maps each element to its equivalence class, is called the canonical ...

5. Equivalence (measure theory) - Wikipedia

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

   Define the two measures on the real line as = [,] () = [,] for all Borel sets. Then and are equivalent, since all sets outside of [,] have and measure zero, and a set inside [,] is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure.

6. Equivalence relation - Wikipedia

   en.wikipedia.org/wiki/Equivalence_relation

   Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals ...

7. Extensionality - Wikipedia

   en.wikipedia.org/wiki/Extensionality

   In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...