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

Famous quotes containing the word examples:

    Histories are more full of examples of the fidelity of dogs than of friends.
    Alexander Pope (1688–1744)

    It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.
    —G.C. (Georg Christoph)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)