Associative Property - Examples

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.,
\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 non-associative.
  • 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:
  • 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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