Editing Associative property (section)

{{Short description|Property of a mathematical operation}}
{{redirect|Associativity }}
{{redirect2|Associative|non-associative|associative and non-associative learning|Learning#Types}}
{{More citations needed|date=June 2009}}
{{Infobox mathematical statement
| name = Associative property
| image = [[File:Associativity of binary operations (without question marks).svg|300px|class=skin-invert-image]]
| caption = A visual graph representing associative operations; <math>(x\circ y)\circ z = x\circ(y\circ z)</math>
| type = [[Principle|Law]], [[rule of replacement]]
| field = {{Plainlist|
* [[Elementary algebra]]
* [[Boolean algebra]]
* [[Set theory]]
* [[Linear algebra]]
* [[Propositional calculus]]
}}
| statement =
| symbolic statement = {{Plainlist|
# Elementary algebra
#: <math>(x \,*\, y) \,*\, z = x \,*\, (y \,*\, z) \forall x,y,z \in S</math>
# Propositional calculus
#:<math>(P \lor (Q \lor R)) \Leftrightarrow ((P \lor Q) \lor R)</math>
#:<math>(P \land (Q \land R)) \Leftrightarrow ((P \land Q) \land R),</math>
}}
}}

In [[mathematics]], the '''associative property'''<ref>
{{cite book
|last=Hungerford
|first=Thomas W.
|year=1974 |edition=1st
|title=Algebra
|page=24
|publisher=[[Springer Science+Business Media|Springer]]
|isbn=978-0387905181
|quote=Definition 1.1 (i) a(bc) = (ab)c for all a, b, c in G.}}</ref> is a property of some [[binary operation]]s that rearranging the [[parentheses]] in an expression will not change the result. In [[propositional logic]], '''associativity''' is a [[Validity (logic)|valid]] [[rule of replacement]] for [[well-formed formula|expressions]] in [[Formal proof|logical proofs]].

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the [[Operation (mathematics)|operations]] are performed does not matter as long as the sequence of the [[operand]]s is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations:

<math display="block">\begin{align}
(2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\
2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 .
\end{align}</math>

Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any [[real number]]s, it can be said that "addition and multiplication of real numbers are associative operations".

Associativity is not the same as [[commutativity]], which addresses whether the order of two operands affects the result. For example, the order does not matter in the multiplication of real numbers, that is, {{math|1={{var|a}} × {{var|b}} = {{var|b}} × {{var|a}}}}, so we say that the multiplication of real numbers is a commutative operation. However, operations such as [[function composition]] and [[matrix multiplication]] are associative, but not (generally) commutative.

Associative operations are abundant in mathematics; in fact, many [[algebraic structure]]s (such as [[semigroup (mathematics)|semigroups]] and [[category (mathematics)|categories]]) explicitly require their binary operations to be associative.

However, many important and interesting operations are non-associative; some examples include [[subtraction]], [[exponentiation]], and the [[vector cross product]]. In contrast to the theoretical properties of real numbers, the addition of [[floating point]] numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.