Codomain

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A codomain in mathematics is the set of "output" values associated with (or mapped to) the domain of "input" arguments in a function. For any given function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon A\rightarrow B} , the set A, on which f is defined (the arguments), is called the domain, and B (the set of possible values) is called the codomain of f. The set of all actual values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lbrace f(x) | x \in A \rbrace} or f(A) is called the range of f.

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, which is in general only a subset of B; in lower-level mathematics education, however, range is often taught as being equivalent to codomain.

Example

Let the function f be a function on the real numbers:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon \mathbb{R}\rightarrow\mathbb{R}}

defined by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\colon\,x\mapsto x^2.}

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,∞):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq f(x)<\infty.}

One could have defined the function g thus:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon\mathbb{R}\rightarrow\mathbb{R}^+_0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\colon\,x\mapsto x^2.}

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. The codomain does not affect whether or not the function is an injection.

See also

de:Wertemenge fr:Ensemble d'arrivée io:Arivey-ensemblo nl:Codomein zh:陪域