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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).
The first use of an equals sign, equivalent to + = in modern notation. From The Whetstone of Witte (1557) by Robert Recorde. Recorde's introduction of =."And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or twin lines of one [the same] length, thus: ==, because no 2 things can be more equal." [5]
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 ).
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 ...
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.
Equality between sets can be expressed in terms of subsets. Two sets are equal if and only if they contain each other: that is, A ⊆ B and B ⊆ A is equivalent to A = B. [30] [8] The empty set is a subset of every set: ∅ ⊆ A. [17] Examples: The set of all humans is a proper subset of the set of all mammals. {1, 3} ⊂ {1, 2, 3, 4}.
The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. [ 2 ] Consider the category C {\displaystyle C} of finite- dimensional real vector spaces , and the category D = M a t ( R ) {\displaystyle D=\mathrm {Mat} (\mathbb {R} )} of all real matrices (the latter ...
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 ...
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