Introducing Operations

    Addition of whole numbers is an example of an operation because for each pair of whole numbers the sum is a unique whole number. Subtraction of integers is an operation because if one integer is subtracted from another the result is an integer. In general, a binary operation on a set is a mapping that assigns to each ordered pair a unique value that is in the set. Operations on sets are the most important concept in modern abstract algebra. This topic defines and illustrates the notions of associative and commutative operations and shows that composition of mappings is an associative binary operation on the set of mappings of a given set.

Definition (Operation) A mapping that assigns to each ordered pair of elements of a uniquely determined element of is a binary operation on

    In order for an operation to be well-defined it is essential that for every ordered pair there must exist an element in that is the image of This property is called closure; or in other words, the set is closed with respect to the operation when precisely, for all

Definition (Cayley Tables) If is a finite set, then specifying an operation by means of a Cayley table is done as follows: Form a square by listing the elements in across the first row and also down the first column. Then fill in every entry in the table from the images of the column and row ordered pairs. Operations on with the same Cayley table are considered equivalent operations on

Example (Operations On Two Element Sets) List all possible operations on the finite set There are of them and they are:

Definition (Associative Law) An operation on a set is said to be associative if it satisfies the condition for all

Definition (Identity Law) An element in a set is an identity for an operation on if for all

Proposition (Unique Identity Element) There is at most one identity element under a binary operation on a set

    Proof. Suppose there is at least one identity element on with respect to the operation and assume that and are identity elements of with respect to Since holds for all replace by thus Similiarily,   holds for all replace by thus Therefore, and so every identity element must be the same, if there is one.  

Definition (Commutative Law) An operation on a set is said to be commutative if for all

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