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Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects).
The diagonal functor: assigns to each object the ordered pair (,) and to each morphism the pair (,). The product X 1 × X 2 {\displaystyle X_{1}\times X_{2}} in C {\displaystyle C} is given by a universal morphism from the functor Δ {\displaystyle \Delta } to the object ( X 1 , X 2 ) {\displaystyle \left(X_{1},X_{2}\right)} in C × C ...
For example, the membership axiom produces a class that may contain elements that are not ordered pairs, while the intersection contains only the ordered pairs of . The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3‑tuples of class B . {\displaystyle B.}
Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category. The surreal numbers are a proper class of objects that have the properties of a field.
It follows that, two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d. Alternatively, an ordered pair can be formally thought of as a set {a,b} with a total order. (The notation (a, b) is also used to denote an open interval on the real number line, but the context should make it clear which meaning is intended.
Given a functor :, let D U be the category where the objects are pairs (x, u) consisting of an object x in C and an object u in the category U(x) and a morphism from (x, u) to (y, v) is a pair consisting of a morphism f: x → y in C and a morphism U(f)(u) → v in U(y).
In any category C, when D is a class of morphisms of C containing identities and closed under composition, the relation 'there exists a D-morphism from X to Y' is a preorder on the class of objects of C. The class Ord of all ordinals is a totally ordered class with the classical ordering of ordinals.
We can use the axiom of extensionality to show that this set C is unique. We call the set C the pair of A and B, and denote it {A,B}. Thus the essence of the axiom is: Any two objects have a pair. The set {A,A} is abbreviated {A}, called the singleton containing A. Note that a singleton is a special case of a pair.