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).
 In arithmetic, addition and multiplication of real numbers are associative; i.e.,

 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.
 Taking the intersection or the union of sets:
 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:
 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
 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.
Read more about this topic: Associative Property
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