Editing Expression (mathematics) (section)

=== Equivalence ===
{{Main|Identity (mathematics)}}

An expression is often used to define a [[Function (mathematics)|function]], or denote [[Function composition|compositions]] of functions, by taking the variables to be [[Argument of a function|arguments]], or inputs, of the function, and assigning the output to be the evaluation of the resulting expression.<ref name="Codd19702">{{cite journal |last1=Codd |first1=Edgar Frank |date=June 1970 |title=A Relational Model of Data for Large Shared Data Banks |url=https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |url-status=live |journal=Communications of the ACM |volume=13 |issue=6 |pages=377–387 |doi=10.1145/362384.362685 |s2cid=207549016 |archive-url=https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |archive-date=2004-09-08 |access-date=2020-04-29 |authorlink=Edgar F. Codd}}</ref> For example, <math>x\mapsto x^2+1</math> and <math>f(x) = x^2 + 1</math>  define the function that associates to each number its [[Square function|square]] plus one. An expression with no variables would define a [[constant function]]. In this way, two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function.<ref>Equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613</ref><ref>Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: https://plato.stanford.edu/entries/algebra/#Laws</ref> The equivalence between two expressions is called an [[Identity (mathematics)|identity]] and is sometimes denoted with <math>\equiv.</math>

For example, in the expression <math display="inline">\sum_{n=1}^{3} (2nx),</math> the variable {{math|''n''}} is bound, and the variable {{math|''x''}} is free. This expression is equivalent to the simpler expression {{math|12 ''x''}}; that is <math display="block">\sum_{n=1}^{3} (2nx)\equiv 12x.</math> The value for {{math|1=''x'' = 3}} is 36, which can be denoted <math display="block">\sum_{n=1}^{3} (2nx)\Big|_{x=3}= 36.</math>