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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional ...
Finally, one can invert the view, switching from functions of orders to orders of functions. Indeed, the functions between two posets P and Q can be ordered via the pointwise order. For two functions f and g, we have f ≤ g if f(x) ≤ g(x) for all elements x of P. This occurs for example in domain theory, where function spaces play an ...
A function that takes a single argument as input, such as () =, is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.
A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not. A bijection [d] A function that is injective and surjective.
[note 3] For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers.
An n-tuple can be formally defined as the image of a function that has the set of the n first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from ordered pairs; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its n th element.
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
The total ordering is defined by considering x and y as sets of ordered pairs (as a function is normally defined): Either x = y, or else the surreal number z = x ∩ y is in the domain of x or the domain of y (or both, but in this case the signs must disagree).