Associativity and Commutativity

It can be shown that for any sets A, B, C,

1.

     $A \cup (B \cup C) = (A \cup B) \cup C$

2.

     $A \cup B = B \cup A$

Result (1) expresses the fact that the union operation ``$\cup$'' is associative and so enables result (1) to be written as $A \cup B \cup C$ without ambiguity, while result (2) states that the union operation is commutative. It can be shown that the operation of set intersection ``$\cap$'' is also both associative and commutative.

Not all set operations are associative and/or commutative. In the case of the set difference operation, as the example demonstrates, the difference operation is not commutative, that is $A - B \not= B - A$. It is also easily verified that the difference operation is not associative either, that is $A - (B - C) \not= (A - B) - C$.

These concepts of associativity and commutativity are of fundamental importance in mathematics. The familiar operations of addition and multiplication over the real numbers are both associative and commutative.