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

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

Let S be any set containing more than two elements. A binary operation @ is defined on S by a@b=b for all a,binS . Discuss the commutativity and associativity of @ on S.

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

A binary operation @ is defined on RR-{-1} by a@b=a+b+ab for all a,binRR-{-1}. (i) Discuss the commutativity and associativity of @ on RR-{-1} . Find the identity element, if exists. (iii) Prove that every element of RR-{-1} is invertible.

(I) Let ** be a binary operation defined by a**b=2a+b-3. Find 3**4. (ii) let ** be a binary operation on RR-{-1} , defined by a**b=(1)/(b+1) for all a,binRR-{-1} Show that ** is neither commutative nor associative. (iii) Let ** be a binary operation on the set QQ of all raional numbers, defined as a**b=(2a-b^(2)) for all a,binQQ . Find 3**5 and 5**3 . Is 3**5=5**3?

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 ** .

Prove that the operation * an Z defined by a*b = a|b| for all a,b in Z is closed under*.

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