Commutativity, Associativity and Distributivity
A binary operation is commutative when , associative when , and distributive over when and .
Definition 2.40 (Commutativity, Associativity and Distributivity of Binary Operations)
We say a binary operation on is
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commutative if , .
e.g. set intersection is commutative
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associative if , .
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distributive over if for another binary operation , , we have
e.g. is distributive over on .
Examples
Set intersection is commutative, and is distributive over on . Composition of functions need not be commutative: for with and with , sends to while sends to . Composition is associative, so its brackets may be dropped without ambiguity.