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(1997), An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions, Cambridge University Press, ISBN978-0-521-59718-0 Forster, Thomas (2003), Logic, Induction and Sets, Cambridge University Press, ISBN978-0-521-53361-4 Mac Lane, Saunders (1998), Categories for the working mathematician (2nd ed.), Springer, ISBN978-0-387-98403-2 Scott, Dana S.; Jech, Thomas J. ^ N.Bourbaki (1954). the codomain of f is R {displaystyle textstyle mathbb {R} } , but f does not map to any negative number. On inspection, h f is not useful. ^ A set of pairs is functional iff no two distinct pairs have the same first component [Bourbaki, op. The codomain is part of a function f if it is defined as described in 1954 by Nicolas Bourbaki, namely a triple (X, Y, F), with F a functional subset of the Cartesian product X Y and X is the set of first components of the pairs in F (the domain). The codomain affects whether a function is a surjection, in that the function is surjective if and only if its codomain equals its image. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. f : R → R {displaystyle fcolon mathbb {R} rightarrow mathbb {R} } . The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. See also.

(2004), Introduction To Set Theory, Discovery Publishing House, ISBN978-81-7141-877-0 Stewart, Ian; Tall, David Orme (1977), The foundations of mathematics, Oxford University Press, ISBN978-0-19-853165-4 . It is the set Y in the notation f: X Y. Function composition therefore is a useful notion only when the codomain of the function on the right side of a composition (not its image, which is a consequence of the function and could be unknown at the level of the composition) is the same as the domain of the function on the left side. Take for example the matrix T given by. Just like all 22 matrices, T represents a member of that set. Contents 1 Examples 2 See also 3 Notes 4 References . .. In general, the image of a function is a subset of its codomain. which represents a linear transformation that maps the point (x, y) to (x, x). Elements de Mathematique,Theorie des Ensembles. f682aff184