Encyclopedia > Associativity

  Article Content

Associativity

In mathematics, a binary operation * on a set S is called associative if for all x, y and z in S, (x * y) * z = x * (y * z).

The most commonly known examples of associativity are addition and multiplication of natural numbers; for example:

  • (7 + 3) + 9 = 7 + (3 + 9), since the expression on the left evaluates to 10 + 9 = 19, which the expression on the right evaluates to 7 + 12 = 19, the same value;
  • (10 × 5) × 3 = 10 × (5 × 3), since the expression on the left evaluates to 50 × 3 = 150, while the expression on the right evalutes to 10 × 15 = 150.

Other examples of associative binary operations include addition and multiplication of real numbers, complex numbers and square matrices; addition of vectors; and intersection and union of sets. Also, 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.

A set with an associative binary operation on it is called a semigroup; monoids and groups are examples of semigroups.

See also Commutativity, Distributive property, Identity element



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Best interests

... - Wikipedia <<Up     Contents Best interests Best interests or best interests of the child is the legal standard used by most courts in ...