Examine whether the operation `@` on `ZZ^(+)` defined by `a@b=|a-b|` for all `a,binZZ^(+)` , is a binary operation on `ZZ^(+)` or not.

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

Determine whether O on Z defined by aOb=a^(b) for all a,b in Z define a binary operation on the given set or not:

Determine whether * on N defined by a*b=a^(b) for all a,b in N define a binary operation on the given set or not:

Determine whether * on N defined by a*b=a+b-2 for all a,b in N define a binary operation on the given set or not:

Show that the operation ** on ZZ , the set of integers, defined by. a**b=a+b-2 for all a,b inZZ (i) is a binary operation: (ii) satisfies commutaitve and associative laws: (iii) Find the identity elemetn in ZZ , (iv) Also find the inverse of an element ainZZ.

Determine whether * on N defined by a*b=a+b-2 for all a ,\ b in N define a binary operation on the given set or not:

Determine whether * on N defined by a*b=a^b for all a ,\ b in N define a binary operation on the given set or not:

Determine whether O on Z defined by a\ O\ b=a^b for all a ,\ b in Z define a binary operation on the given set or not:

Find the identity element of the binary operation ** on ZZ defined by a**b=a+b+1 for all a,binZZ .

Check for commutativity and associativity @ on ZZ defined by a@b=a|b| for all a,binZZ .