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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, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from A to B such that for every element y of B, there is exactly one element x of A with f(x) = y. [1] Equinumerous sets are said to have the same cardinality (number of ...
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 ...
For example, 7 ≥ 5 but not 5 ≥ 7. The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common ...
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 ).
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.
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 ...
Forming a partial setoid from a type and a PER is analogous to forming subsets and quotients in classical set-theoretic mathematics. The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence, i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reflexive ...
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