Substitution ○

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Substitution ○｜Definition｜1st｜20251119205401-00-⌔

Substitution (logic) - Wikipedia#Algebra

Algebra

Substitution is a basic operation in algebra, in particular in computer algebra.¹²

A common case of substitution involves polynomials, where substitution of a numerical value (or another expression) 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

• ✤ $P(X)=X^5-3X^2+5X-17$

so that substitution for X can be designated by replacement inside “P (X)”, say

• ✤ $P(2)=13$

or

• ✤ $P(X+1)=X^5+5X^4+10X^3+7X^2+4X-14.$

Substitution can also be applied to other kinds of formal objects built from symbols, for instance elements of free groups. In order for substitution to be defined, one needs an algebraic structure with an appropriate universal property, that asserts the existence of unique homomorphisms that send indeterminates to specific values; the substitution then amounts to finding the image of an element under such a homomorphism.

Printed 2026-06-28.

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Footnotes

1. Margret H. Hoft; Hartmut F.W. Hoft (6 November 2002). Computing with Mathematica. Elsevier. ISBN 978-0-08-048855-4. ↩

2. Andre Heck (6 December 2012). Introduction to Maple. Springer Science & Business Media. ISBN 978-1-4684-0484-5. substitution. ↩

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