In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the operation of substitution can be applied in various contexts involving formal objects containing symbols (often called
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s or
indeterminates); the operation consists of systematically replacing occurrences of some symbol by a given value.
Substitution is a basic operation of
computer algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
.
It is generally called "subs" or "subst" in
computer algebra system
A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The de ...
s.
A common case of substitution involves
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, where substitution of a numerical value for the indeterminate of a (univariate) polynomial amounts to evaluating the polynomial at that value. Indeed, this operation occurs so frequently that the notation for polynomials is often adapted to it; instead of designating a polynomial by a name like ''P'', as one would do for other mathematical objects, one could define
:
so that substitution for ''X'' can be designated by replacement inside "''P''(''X'')", say
:
or
:
Substitution can however also be applied to other kinds of formal objects built from symbols, for instance elements of
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
s. In order for substitution to be defined, one needs an algebraic structure with an appropriate
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
, that asserts the existence of unique homomorphisms that send indeterminates to specific values; the substitution then amounts to finding the image under such a homomorphism.
Substitution is related to, but not identical to,
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
; it is also closely related to ''β''-reduction in
lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation ...
. In contrast to these notions, however, the accent in algebra is on the preservation of algebraic structure by the substitution operation, the fact that substitution gives a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
for the structure at hand (in the case of polynomials, the
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
structure).
See also
*
Substitution (logic) Substitution is a fundamental concept in logic.
A substitution is a syntactic transformation on formal expressions.
To apply a substitution to an expression means to consistently replace its variable, or placeholder, symbols by other expressions.
...
— about a formal treatment of substitution
*
Integration by substitution
In calculus, integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation, and can ...
*
Trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities ...
References
Algebra
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