Definition – Operation (on a set) .

Definition

Let $A$ be a set. Consider the set of ordered pairs resulting from the cross product of $A$ with itself, $A\times{A}$ . An operation on the set $A$ is a rule which assigns a unique element of the set $A$ , denoted by $a_1 * a_2$ , to every element $(a_1, a_2)\in{A\times{A}}$ .

Properties of operations

Note that the definition states that the element $a_1 * a_2$ is a member of the set $A$ . If this is not the case, then the operation $*$ is not an operation on $A$ , as per the definition.

Also, if the above property is true, we say that the set $A$ is closed under the operation $*$ .