In mathematics, the associative property is a property of some binary operations. 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, rearranging the parentheses in such an expression will not change its value. Consider, for instance, the following equations:
 (5+2)+1=5+(2+1)=8 \,
 5\times(5\times3)=(5\times5)\times3=75 \,
Consider the first equation. Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the value of the expression was not altered. Since this holds true when performing addition on any real numbers, we say that "addition of real numbers is an associative operation."
Associativity is not to be confused with commutativity. Commutativity justifies changing the order or sequence of the operands within an expression while associativity does not. For example,
 (5+2)+1=5+(2+1) \,
is an example of associativity because the parentheses were changed (and consequently the order of operations during evaluation) while the operands 5, 2, and 1 appeared in exactly the same order from left to right in the expression. In contrast,
 (5+2)+1=(2+5)+1 \,
is an example of commutativity, not associativity, because the operand sequence changed when the 2 and 5 switched places.
Associative operations are abundant in mathematics; in fact, many algebraic structures (such as semigroups and categories) explicitly require their binary operations to be associative.
However, many important and interesting operations are nonassociative; one common example would be the vector cross product.
Definition
Formally, a binary operation *\!\!\! on a set S is called associative if it satisfies the associative law:
 (x * y) * z=x * (y * z)\qquad\mbox{for all }x,y,z\in S.
 Using * to denote a binary operation performed on a set
 (xy)z=x(yz) = xyz \qquad\mbox{for all }x,y,z\in S.
 An example of multiplicative associativity
The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of *\!\!\! operations. Thus, when *\!\!\! is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:
 xyz,
However, it is important to remember that changing the order of operations does not involve or permit moving the operands around within the expression; the sequence of operands is always unchanged.
The associative law can also be expressed in functional notation thus : f(f(x,y),z) = f(x,f(y,z)).
Associativity can be generalized to nary operations. Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. Nary associativity is a string of length n+(n1) with any n adjacent elements bracketed.^{[1]}
Examples
Some examples of associative operations include the following.
 The concatenation of the three strings
"hello" , " " , "world" can be computed by concatenating the first two strings (giving "hello " ) and appending the third string ("world" ), or by joining the second and third string (giving " world" ) and concatenating the first string ("hello" ) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).

 \left. \begin{matrix} (x+y)+z=x+(y+z)=x+y+z\quad \\ (x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \, \end{matrix} \right\} \mbox{for all }x,y,z\in\mathbb{R}.
 Because of associativity, the grouping parentheses can be omitted without ambiguity.
 Addition and multiplication of complex numbers and quaternions is associative. Addition of octonions is also associative, but multiplication of octonions is nonassociative.
 The greatest common divisor and least common multiple functions act associatively.

 \left. \begin{matrix} \operatorname{gcd}(\operatorname{gcd}(x,y),z)= \operatorname{gcd}(x,\operatorname{gcd}(y,z))= \operatorname{gcd}(x,y,z)\ \quad \\ \operatorname{lcm}(\operatorname{lcm}(x,y),z)= \operatorname{lcm}(x,\operatorname{lcm}(y,z))= \operatorname{lcm}(x,y,z)\quad \end{matrix} \right\}\mbox{ for all }x,y,z\in\mathbb{Z}.
 Taking the intersection or the union of sets:

 \left. \begin{matrix} (A\cap B)\cap C=A\cap(B\cap C)=A\cap B\cap C\quad \\ (A\cup B)\cup C=A\cup(B\cup C)=A\cup B\cup C\quad \end{matrix} \right\}\mbox{for all sets }A,B,C.
 If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

 (f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h\qquad\mbox{for all }f,g,h\in S.
 Slightly more generally, given four sets M, N, P and Q, with h: M to N, g: N to P, and f: P to Q, then

 (f\circ g)\circ h=f\circ(g\circ h)=f\circ g\circ h
 as before. In short, composition of maps is always associative.
 Consider a set with three elements, A, B, and C. The following operation:

A 
B 
C 
A 
A 
A 
A 
B 
A 
B 
C 
C 
A 
A 
A 
is associative. Thus, for example, A(BC)=(AB)C. This mapping is not commutative.
 Because matrices represent linear transformation functions, with matrix multiplication representing functional composition, one can immediately conclude that matrix multiplication is associative.
Propositional logic
Rule of replacement
In standard truthfunctional propositional logic, association^{[2]}^{[3]}, or associativity^{[4]} are two valid rules of replacement. The rules allow one to move parentheses in logical expressions in logical proofs. The rules are:
 (P \or (Q \or R)) \Leftrightarrow ((P \or Q) \or R)
and
 (P \and (Q \and R)) \Leftrightarrow ((P \and Q) \and R)
Where "\Leftrightarrow" is a metalogical symbol representing "can be replaced in a proof with."
Truth functional connectives
Associativity is a property of some logical connectives of truthfunctional propositional logic. The following logical equivalences demonstrate that associativity is a property of particular connectives. The following are truthfunctional tautologies.
Associativity of disjunction:
 (P \or (Q \or R)) \leftrightarrow ((P \or Q) \or R)
 ((P \or Q) \or R) \leftrightarrow (P \or (Q \or R))
Associativity of conjunction:
 ((P \and Q) \and R) \leftrightarrow (P \and (Q \and R))
 (P \and (Q \and R)) \leftrightarrow ((P \and Q) \and R)
Associativity of equivalence:
 ((P \leftrightarrow Q) \leftrightarrow R) \leftrightarrow (P \leftrightarrow (Q \leftrightarrow R))
 (P \leftrightarrow (Q \leftrightarrow R)) \leftrightarrow ((P \leftrightarrow Q) \leftrightarrow R)
Nonassociativity
A binary operation * on a set S that does not satisfy the associative law is called nonassociative. Symbolically,
 (x*y)*z\ne x*(y*z)\qquad\mbox{for some }x,y,z\in S.
For such an operation the order of evaluation does matter. For example:
 (53)2 \, \ne \, 5(32)
 (4/2)/2 \, \ne \, 4/(2/2)
 2^{(1^2)} \, \ne \, (2^1)^2
Also note that infinite sums are not generally associative, for example:
 (11)+(11)+(11)+(11)+(11)+(11)+\dots \, = \, 0
whereas
 1+(1+1)+(1+1)+(1+1)+(1+1)+(1+1)+(1+\dots \, = \, 1
The study of nonassociative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within nonassociative algebra that has grown very large is that of Lie algebras. There the associative law is replaced by the Jacobi identity. Lie algebras abstract the essential nature of infinitesimal transformations, and have become ubiquitous in mathematics. They are an example of nonassociative algebras.
There are other specific types of nonassociative structures that have been studied in depth. They tend to come from some specific applications. Some of these arise in combinatorial mathematics. Other examples: Quasigroup, Quasifield, Nonassociative ring.
Notation for nonassociative operations
In general, parentheses must be used to indicate the order of evaluation if a nonassociative operation appears more than once in an expression. However, mathematicians agree on a particular order of evaluation for several common nonassociative operations. This is simply a notational convention to avoid parentheses.
A leftassociative operation is a nonassociative operation that is conventionally evaluated from left to right, i.e.,
 \left. \begin{matrix} x*y*z=(x*y)*z\qquad\qquad\quad\, \\ w*x*y*z=((w*x)*y)*z\quad \\ \mbox{etc.}\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end{matrix} \right\} \mbox{for all }w,x,y,z\in S
while a rightassociative operation is conventionally evaluated from right to left:
 \left. \begin{matrix} x*y*z=x*(y*z)\qquad\qquad\quad\, \\ w*x*y*z=w*(x*(y*z))\quad \\ \mbox{etc.}\qquad\qquad\qquad\qquad\qquad\qquad\ \ \, \end{matrix} \right\} \mbox{for all }w,x,y,z\in S
Both leftassociative and rightassociative operations occur. Leftassociative operations include the following:
 Subtraction and division of real numbers:

 xyz=(xy)z\qquad\mbox{for all }x,y,z\in\mathbb{R};

 x/y/z=(x/y)/z\qquad\qquad\quad\mbox{for all }x,y,z\in\mathbb{R}\mbox{ with }y\ne0,z\ne0.

 (f \, x \, y) = ((f \, x) \, y)
 This notation can be motivated by the currying isomorphism.
Rightassociative operations include the following:

 x^{y^z}=x^{(y^z)}.\,
 The reason exponentiation is rightassociative is that a repeated leftassociative exponentiation operation would be less useful. Multiple appearances could (and would) be rewritten with multiplication:

 (x^y)^z=x^{(yz)}.\,

 \mathbb{Z} \rarr \mathbb{Z} \rarr \mathbb{Z} = \mathbb{Z} \rarr (\mathbb{Z} \rarr \mathbb{Z})

 x \mapsto y \mapsto x  y = x \mapsto (y \mapsto x  y)
 Using rightassociative notation for these operations can be motivated by the CurryHoward correspondence and by the currying isomorphism.
Nonassociative operations for which no conventional evaluation order is defined include the following.
 Taking the Cross product of three vectors:

 \vec a \times (\vec b \times \vec c) \neq (\vec a \times \vec b ) \times \vec c \qquad \mbox{ for some } \vec a,\vec b,\vec c \in \mathbb{R}^3
 Taking the pairwise average of real numbers:

 {(x+y)/2+z\over2}\ne{x+(y+z)/2\over2} \qquad \mbox{for all }x,y,z\in\mathbb{R} \mbox{ with }x\ne z.
 Taking the relative complement of sets (A\backslash B)\backslash C is not the same as A\backslash (B\backslash C). (Compare material nonimplication in logic.)
See also
 Light's associativity test
 A semigroup is a set with a closed associative binary operation.

Commutativity and distributivity are two other frequently discussed properties of binary operations.
 Power associativity and alternativity are weak forms of associativity.
References
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