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The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change.
In mathematics, the associative property [1] is a property of some binary operations that means 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 .
An operation that is mathematically associative, by definition requires no notational associativity. (For example, addition has the associative property, therefore it does not have to be either left associative or right associative.) An operation that is not mathematically associative, however, must be notationally left-, right-, or non ...
In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A.This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K).
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
For the integers and the operation addition +, denoted (, +), the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer has an additive inverse, , and the addition operation is commutative since + = + for any two integers and .
The property of two matrices commuting is not transitive: A matrix may commute with both and , and still and do not commute with each other. As an example, the identity matrix commutes with all matrices, which between them do not all commute.
The first three examples above are commutative and all of the above examples are associative. On the set of real numbers R {\displaystyle \mathbb {R} } , subtraction , that is, f ( a , b ) = a − b {\displaystyle f(a,b)=a-b} , is a binary operation which is not commutative since, in general, a − b ≠ b − a {\displaystyle a-b\neq b-a} .