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 nonidentity 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 nonempty 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 nonempty 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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Properties Of Operations
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/properties-of-operations.html