Is aob = a is associative in S ? : Group Theory

 

Problem :

 Let S be a non-empty set and ‘ o ‘ be an operation on S defined by aob = a for a,b 𝟄 S.

Determine whether  o is commutative and associative in S?

Solution :

 Let S be a non-empty set and ‘ o ‘ be an operation on S defined by aob = a for a,b 𝟄 S.

Checking for associative :

    Let a,b,c 𝟄 S.

   Now (aob)oc = aoc        | since aob = a |

                          =  a            | since aoc = a |

      ∴ (aob)oc = a

 Also  ao(boc) = aob          | since boc = b |

                         = a                | since aob = a |

Since (aob)oc = ao(boc) ∀ a,b,c 𝟄 S.

  ∴ ‘o’ is associative in S.

Checking for commutative :

    Let a,b 𝟄 S.

 Now  aob = a   ( by the definition )

 Also boa = b     ( by the definition)

Since aob ≠ boa , ‘o’ is not commutative in S.

           *** Hence Solved ***

 * What is groupoid

 * What is monoid

 * What is abelian group

 * Uniqueness of identity element 

 *  Is aob = a^b is a binary operation on N?

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