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This article is about associativity in mathematics. For associativity in central processor unit memory cache architecture see CPU cache. In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are required for an associative operation. Consider for instance the equation (5+2)+1 = 5+(2+1) Adding 5 and 2 gives 7, and adding 1 gives an end result of 8 for the left hand side. To evaluate the right hand side, we start with adding 2 and 1 giving 3, and then add 5 and 3 to get 8, again. So the equation holds true. In fact, it holds true for all real numbers, not just for 5, 2 and 1. We say that "addition of real numbers is an associative operation". Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; 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: 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 therefore be left unspecified without causing ambiguity, by omitting the parentheses and writing simply: Examples Some examples of associative operations include the following. left. egin (x+y)+z=x+(y+z)=x+y+zquad \ (x,y)z=x(y,z)=x,y,zqquadqquadqquadquad , end ight} mboxx,y,zinmathbb. left. egin operatorname(operatorname(x,y),z)= operatorname(x,operatorname(y,z))= operatorname(x,y,z) quad \ operatorname(operatorname(x,y),z)= operatorname(x,operatorname(y,z))= operatorname(x,y,z)quad end ight}mboxx,y,zinmathbb. left. egin (Acap B)cap C=Acap(Bcap C)=Acap Bcap Cquad \ (Acup B)cup C=Acup(Bcup C)=Acup Bcup Cquad end ight}mboxA,B,C. as before. In short, composition of maps is always associative. Non-associativity A binary operation on a set S that does not satisfy the associative law is called non-associative. Symbolically, e x egin (5-3)-2 e 5-(3-2)quad \ (4/2)/2 e 4/(2/2)qquadqquad \ 2^ e (2^1)^2.quadqquadqquad end In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression. However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This is simply a syntactical convention to avoid parentheses. A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e., left. egin x w mboxqquadqquadqquadqquadqquadqquad , end ight} mboxw,x,y,zin S while a right-associative operation is conventionally evaluated from right to left: left. egin x w mboxqquadqquadqquadqquadqquadqquad , end ight} mboxw,x,y,zin S Both left-associative and right-associative operations occur; examples are given below. More examples Left-associative operations include the following. e0,z e0. Right-associative operations include the following. The reason exponentiation is right-associative is that a repeated left-associative exponentiation operation would be less useful. Multiple appearances could (and would) be rewritten with multiplication: Non-associative operations for which no conventional evaluation order is defined include the following. e eqquadmboxx,y,zinmathbb. e Aackslash (Backslash C)qquadmboxA,B,C. The green part in the left Venn diagram represents (A'B)C. The green part in the right Venn diagram represents A(B'C) See also | ||||||||
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