Prove that the binary operation `**` on `RR` defined by
`a**b=a+b+ab` for all `a,binRR`
is commutative and associative.

Prove that the binary operation @ defined on QQ by a@b=a-b+ab for all a,b in QQ is neither commutative nor associative.

The binary operation ** defined on NN by a**b=a+b+ab for all a,binNN is--

Let ** and @ be two binary operations on RR defined as, a**b=|a-b|anda@b=a for all a,binRR . Examine the commutativity and associativity of ** and @ on RR . Show also that ** is distributative over @ but @ is not distributive over ** .

Show that identity of the binary operation ** on RR defined by a**b=|a+b| for all a,binRR, does not exist.

Prove that the operation ** on ZZ defined by a**b=a|b| for all a,binZZ is a binary operation

Prove that the binary operation O defined by a o b = a-b+ab, AA a,b in Q is not commutative and associative.

Prove that the binary operation o defined by a o b = |a|+|b|, AA a,b in R is commutative and associatie on R.

The binary operation * define on N by a*b = a+b+ab for all a,binN is

Let * be a binary operation on Zdefined by a*b=a+b- 15 for a,binZ . Show that *is commutative

Prove that the identity element of the binary opeartion ** on RR defined by a**b = min. (a,b) for all a,binRR , does not exist.