Associative property
property of binary operations allowing sequences of operations to be regrouped without changing their value
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands 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:
(
2
+
3
)
+
4
=
2
+
(
3
+
4
)
=
9
2
×
(
3
×
4
)
=
(
2
×
3
)
×
4
=
24.
{\displaystyle {\begin{aligned}(2+3)+4&=2+(3+4)=9\,\\2\times (3\times 4)&=(2\times 3)\times 4=24.\end{aligned}}}
Even though the parentheses were rearranged on each line, the values of the expressions were not altered. This is always true when performing additions and multiplications of real numbers, since 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, a × b = b × 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.
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