Operations

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.

Example (Operation) The following are examples of operations defined on a set.

    (i) Addition is a closed binary operation on because the sum of two positive integer is a positive integer. Subtraction is not a closed binary operation on Although both 5 and 7 are in is not positive and so not in
    
    (ii) Subtraction is a closed binary operation on because the difference of any two integers is again an integer.
    
    (iii) Multiplication is a closed binary operation on the sets and
    
    (iv) Define by is a closed operation on or but not on since which is not an integer.
    
    (v) Define by is a closed binary operation on the sets and
    

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

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

The following proposition explains why we can say "the identity" and instead of "an identity".

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 Similarly,   holds for all replace by thus Therefore, and so every identity element must be the same, if there is one.  

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

Example (Commutative but not Associative) Let be our set and define by for all Then for any and



and so is commutative. However,





but for all and so which means is not associative.

Properties of Operations

Proposition (Properties of Operations) Let be an operation on

    (i)  If has an identity element, then it suffices to check the associativity condition for the non-identity elements of
    
    (ii): Assume that has an identity element and for all then is commutative and associative.
    
    (ii) If is associative, then  
    
    Proof.    (i): Let If is the identity element, then for all If is the identity element, then for all   If is the identity element, then for all Therefore, to check for associativity it suffices to check that where none of is the identity element.
    (ii): Let be the identity element, then and so is commutative. So then showing that is associative.
    (iii): By definition of associative,

Proposition (Composition Operations) Let denote any non-empty set and the set of all mappings from to

    (i) Composition is an associative operation on with identity element
    
    (ii) Composition is an associative operation on the set of all invertible mappings in with identity
    
    (iii): More generally, if and denote any non-empty sets and  if and then
    
    Proof.    (i): Let and Then Since for all the mapping is an identity element on with respect to the operation of composition.
    (ii): Composition on the set of all invertible mappings in is indeed an operation because if and are invertible then so is By (i) this operation is associative and is the identity.
    (iii): If then   

Proposition (Unique Inverse) Every invertible mapping has a unique inverse.

    Proof. Let and assume that both and are inverses of Then and Then by the Composition Operation Proposition, which yields, for all Therefore, and inverses are unique.  

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