Associativity of binary operations

Discuss the commutativity and associativity of binary operation* defined on Q by the rule a*b=a-b+ab for all a,b in Q

Discuss the commutativity and associativity of the binary operation * on R defined by a*b=(ab)/(4) for all a,b in R

Discuss the commutativity and associativity of the binary operation * on R defined by a*b=a-b+ab for all a,b in R, where on RHS we have usual addition,subtraction and multiplication of real numbers.

Commutativity of binary operations

Distributivity of binary operations

Define an associative binary operation on a set.

Number of binary operations

Show that addition and multiplication are associative binary operation on R.But subtraction is not asociative on R.Division is not associative on R^(*).

Let * be an associative binary operation on a set S with the identity element e in S. Then. the inverse of an invertible element is unique.

Let * be an associative binary operation on a set S and a be an invertible element of S. Then; inverse of a^(^^)-1 is a.