What does codomain mean?
In mathematics, the codomain of a function is the set of all values the function is defined or declared to be able to produce. Every function maps elements from one set, called the domain, to another, called the codomain. Crucially, the codomain need not coincide with the function's range: the range is the subset of the codomain containing only the values actually attained. For example, the squaring function from real numbers to real numbers has all reals as its codomain, but its range consists solely of non-negative numbers. The distinction matters in fields such as linear algebra, topology, and category theory, where specifying whether a function is surjective — meaning its range equals its codomain — depends on this precise terminology. The word derives from 'domain', with the prefix 'co-' indicating a paired or dual role, and it entered mathematical English in the twentieth century alongside modern set-theoretic foundations.
nounThe set of all values that a function could possibly output, as specified in its definition. It contains the range (the set of values the function actually attains) and may be strictly larger than it.
- The specified set of possible outputs of a function, within which its range lies.
"The codomain of the function f(x) = x² is the set of real numbers, even though its range is only the non-negative reals."
"Every function from ℝ to ℝ has ℝ as its codomain by convention."
"When defining a surjective map, mathematicians often adjust the codomain so that the range fills it entirely."
Standard pluralization; used when comparing functions or discussing multiple mappings.
"The two functions have different codomains, so they cannot be composed directly."
A function's codomain can be bigger than what it actually produces — like promising to cook any dish in the cookbook but only ever making soup.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of codomain
The word 'codomain' combines 'co-', a prefix of Latin origin meaning 'together' or 'jointly', with 'domain', from Latin 'dominium' ('lordship, property'), via Old French. The compound arose within twentieth-century mathematics as set theory standardized the language of functions, pairing 'codomain' with 'domain' to name the two sets between which a function operates.
How codomain is actually used
A technical term used almost exclusively in mathematics, particularly set theory, algebra, and category theory. In informal usage, students sometimes conflate codomain with range; careful writers reserve 'range' for the actual image of the function.
Easily confused with codomain
The codomain is the full set a function is declared to output into, while the range is the narrower subset of that set which the function actually reaches.
The domain is the set of inputs a function accepts, whereas the codomain is the set into which its outputs fall.