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If −1 < x < 1 there are two possible values of y, one positive and one negative. For x = ± 1, these two values become both equal to 0. Otherwise, there is no possible value of y. This means that the equation defines two implicit functions with domain [−1, 1] and respective codomains [0, +∞) and (−∞, 0].
A function may be defined as a binary relation that meets additional constraints. [3] Binary relations are also heavily used in computer science . A binary relation over sets X {\displaystyle X} and Y {\displaystyle Y} is an element of the power set of X × Y . {\displaystyle X\times Y.}
In mathematics, a finitary relation over a sequence of sets X 1, ..., X n is a subset of the Cartesian product X 1 × ... × X n; that is, it is a set of n-tuples (x 1, ..., x n), each being a sequence of elements x i in the corresponding X i. [1] [2] [3] Typically, the relation describes a possible connection between the elements of an n-tuple.
If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t. Function defined on finite sets which change the order of their elements such as permutations can be composed on the same set, this being composition of permutations.
What is Mathematics? An Elementary Approach to Ideas and Methods, book review by Brian E. Blank, Notices of the American Mathematical Society 48, #11 (December 2001), pp. 1325–1330; What is Mathematics?, book review by Leonard Gillman, The American Mathematical Monthly 105, #5 (May 1998), pp. 485–488.
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain).
Analytic continuation of natural logarithm (imaginary part) Analytic continuation is a technique to extend the domain of a given analytic function.Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
The problem of maximizing a non-negative submodular function admits a 1/2 approximation algorithm. [ 19 ] [ 20 ] Computing the maximum cut of a graph is a special case of this problem. The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a 1 − 1 / e {\displaystyle 1-1/e} approximation algorithm.