different between surjective vs codomain

English

Pronunciation

• Rhymes: -?kt?v

Adjective

surjective (not comparable)

1. (mathematics) of, relating to, or being a surjection
   • 1974, Thomas W. Hungerford, Algebra, Springer, page 5,

     A function ${\displaystyle \textstyle f}$ is surjective (or onto) provided ${\displaystyle \textstyle f(A)=B}$; in other words,

     for each ${\displaystyle \textstyle b\in B,b=f(a)}$ for some ${\displaystyle \textstyle a\in A}$.

     A function ${\displaystyle \textstyle f}$ is said to be bijective (or a bijection or a one-to-one correspondence) if it is both injective and surjective.

   • 2010, Tullio Ceccherini-Silberstein, Michel Coornaert, Cellular Automata and Groups, Springer, page 133,

     The Garden of Eden Theorem (Theorem 5.3.1) implies that every surjective cellular automaton with finite alphabet over an amenable group is necessarily pre-injective. In this section, we give an example of a surjective but not pre-injective cellular automaton with finite alphabet over the free group ${\displaystyle \textstyle F_{2}}$.

   • 2011, Ethan D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, Springer, 2nd Edition, page 156,

     This function is surjective and injective, and hence bijective.

   Synonym: onto

Derived terms

Related terms

Translations

See also

• bijective
• injective

French

Pronunciation

• IPA^(key): /sy?.??k.tiv/
• Homophone: surjectives

Adjective

surjective

1. feminine singular of surjectif

English

Etymology

co- +? domain

Pronunciation

• (US) IPA^(key): /?ko?.do??me?n/

Noun

codomain (plural codomains)

1. (mathematics, mathematical analysis) The target set into which a function is formally defined to map elements of its domain; the set denoted Y in the notation f : X ? Y.
   • 1994, Richard A. Holmgren, A First Course in Discrete Dynamical Systems, Springer, page 11,

     Definition 2.5. A function is onto if each element of the codomain has at least one element of the domain assigned to it. In other words, a function is onto if the range equals the codomain.

   • 2006, Robert L. Causey, Logic, Sets, and Recursion, 2nd Edition, Jones & Bartlett Learning, page 192,

     Once we have described ${\displaystyle f}$ as a function from ${\displaystyle A}$ to ${\displaystyle B}$, by convention we will call ${\displaystyle B}$ the codomain, even though other sets, of which ${\displaystyle B}$ is a subset, could have been used. […] If ${\displaystyle y}$ is an element of the codomain, then ${\displaystyle y\in {\mathit {Img}}(f,A)}$ iff there is some ${\displaystyle x}$ in the domain such that ${\displaystyle f}$ maps ${\displaystyle x}$ to ${\displaystyle y}$.

   • 2017, Alan Garfinkel, Jane Shevtsov, Yina Guo, Modeling Life: The Mathematics of Biological Systems, Springer, page 12,

     For example, the codomain of ${\displaystyle g(X)=X^{3}}$ consists of all real numbers. A function links each element in its domain to some element in its codomain. Each domain element is linked to exactly one codomain element.

Usage notes

The codomain always contains the image of the function (the actual set of points to which points of the domain are mapped), and can be larger if the function is not surjective.

The term range is often synonymous with codomain, but can also be used as a synonym for image.

Synonyms

• (target set of a function): range

Antonyms

• (target set of a function): domain

Translations

Further reading

• Domain of a function on Wikipedia.Wikipedia
• Image (mathematics) on Wikipedia.Wikipedia
• Range (mathematics) on Wikipedia.Wikipedia
• Injective function on Wikipedia.Wikipedia
• Surjective function on Wikipedia.Wikipedia
• Bijection on Wikipedia.Wikipedia
• Codomain on Wolfram MathWorld

Anagrams

• monoacid