codomain: meaning, definition, pronunciation and examples
C2Academic, Technical
Quick answer
What does “codomain” mean?
In mathematics, the set into which all outputs of a function are constrained to fall.
Audio
Pronunciation
Definition
Meaning and Definition
In mathematics, the set into which all outputs of a function are constrained to fall.
More generally, in a mapping from a domain to a codomain, it is the target set that contains all possible values the function can produce, though not all elements of the codomain need be actual outputs.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Both regions use the term identically within mathematical discourse.
Connotations
Purely technical and precise. No regional connotations.
Frequency
Identically low frequency in both regions, restricted to academic and technical contexts.
Grammar
How to Use “codomain” in a Sentence
[function] has/takes [domain] as its domain and [codomain] as its codomain.The codomain of [function] is [set].Vocabulary
Collocations
Examples
Examples of “codomain” in a Sentence
adjective
British English
- The codomain set must be specified.
- A codomain restriction can change a function's properties.
American English
- The codomain element was not reached.
- We need a larger codomain for this mapping.
Usage
Meaning in Context
Business
Not used.
Academic
Standard term in pure mathematics, computer science, and related fields when defining functions and mappings.
Everyday
Virtually never used.
Technical
Essential concept in formal logic, set theory, category theory, and programming language theory.
Vocabulary
Synonyms of “codomain”
Neutral
Weak
Vocabulary
Antonyms of “codomain”
Watch out
Common Mistakes When Using “codomain”
- Confusing 'codomain' with 'range' or 'image'. Using the terms interchangeably.
- Assuming every element of the codomain has a corresponding input in the domain (this describes a surjective/onto function, not the definition of codomain).
FAQ
Frequently Asked Questions
No. The range is the set of actual outputs. The codomain is the set of all possible outputs. The range is always a subset of (or equal to) the codomain. They are equal only for a surjective (onto) function.
It provides a clear, upfront constraint on what type of outputs are acceptable. It allows us to discuss properties like surjectivity (whether the function 'hits' every element of the codomain) and to define function composition correctly.
In modern set-theoretic definitions, the codomain is an intrinsic part of a function's definition. Changing the codomain technically creates a different function, even if the mapping rule is the same. For example, f: R -> R and g: R -> [0, ∞) with f(x)=x^2 and g(x)=x^2 are considered different functions.
A vending machine (function). DOMAIN: The set of buttons you can press (inputs). CODOMAIN: The set of all snack types the machine stocks. RANGE: The specific snacks that are currently not sold out (actual outputs you can get).
Codomain is usually academic, technical in register.
Codomain: in British English it is pronounced /ˈkəʊdəmeɪn/, and in American English it is pronounced /ˈkoʊdoʊˌmeɪn/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of CODEPENDENT and DOMAIN. A function is "co-dependent" on its DOMAIN (input set) and its CODOMAIN (the set where it promises its outputs will live).
Conceptual Metaphor
A POSTAL SYSTEM: The domain is the set of all letters to be sent. The codomain is the set of all possible addresses (towns) in the country. The range is the specific set of towns that actually receive letters.
Practice
Quiz
What is the relationship between the range (image) of a function and its codomain?