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Encyclopedia > Codomain

A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "inputs" in a function. For any given function , the set B is called the codomain of f. X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. The codomain is not to be confused with the range f(A), which is in general only a subset of B; in lower-level mathematics education, however, range is often taught as being equivalent to codomain. Mathematics is the study of quantity, structure, space and change. ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... Information processing In information processing, output is the process of transmitting information (verb usage). ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Domain has several meanings: some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ... Information processing In information processing, input is the process of receiving information from an object. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a set can be thought of as any well-defined collection of things considered as a whole. ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all values produced by a function. ... In mathematics, the range of a function is the set of all values produced by a function. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... Mathematics education is the study of practices and methods of both the teaching and learning of mathematics. ...


Example

Let the function f be a function on the real numbers: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

defined by

The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R+—non-negative reals, i.e. the interval [0,∞): A negative number is a number that is less than zero, such as −3. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

One could have defined the function g thus:

While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.


The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...


See also


  Results from FactBites:
 
Codomain - Wikipedia, the free encyclopedia (244 words)
A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "input" arguments in a function.
The codomain is not to be confused with the range, which is in general only a subset of B; in lower-level mathematics education, however, range is often taught as being equivalent to codomain.
The codomain does not affect whether or not the function is an injection.
Range (mathematics) - Wikipedia, the free encyclopedia (239 words)
The range is a subset of the codomain, but is not necessarily equal to the codomain, since there may be elements of the codomain which are not elements of the range.
Another way to think about this is to consider the codomain to be the set of all possible output values, while the range is the set of all actual outputs.
The codomain of f is R, and f takes all nonnegative values but never takes negative values, and thus the range is in fact the set R
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