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Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry ), a notion (for example, ellipse or minimal surface ) may have more than one definition.
In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. [1] Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS).
Every element of is a member of the equivalence class []. Every two equivalence classes [] and [] are either equal or disjoint.Therefore, the set of all equivalence classes of forms a partition of : every element of belongs to one and only one equivalence class. [4]
In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relation (although not the only one; other common choices include ~ and ≈). [ 7 ] [ 8 ] Particularly, in geometry , it may be used either to show that two figures are congruent or that they are identical. [ 9 ]
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics.
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which ...
Equality (mathematics) Equality operator; Equipollence (geometry) Equivalence (measure theory) Equivalence class; Equivalence of categories; Equivalence of metrics; Equivalence relation; Equivalence test; Equivalent definitions of mathematical structures; Equivalent infinitesimal; Equivalent latitude; Exponentially equivalent measures ...