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For example: "An even number is an integer which is divisible by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -8..." In logic, the extension of a predicate is the set of all things for which the predicate is true. [49]
A functor F : C → D yields an equivalence of categories if and only if it is simultaneously: . full, i.e. for any two objects c 1 and c 2 of C, the map Hom C (c 1,c 2) → Hom D (Fc 1,Fc 2) induced by F is surjective;
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).
In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
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, 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.
As another example, the statement "the solution to an indefinite integral is sin(x), up to addition of a constant" tacitly employs the equivalence relation R between functions, defined by fRg if the difference f−g is a constant function, and means that the solution and the function sin(x) are equal up to this R.
For example it is well known that in univalent foundations, the univalence axiom implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational content must be preserved.
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