Determine whether or not the definition of * On `R` , define by `a*b=a b^2` gives a binary operation. In the event that * is not a binary operation give justification of this. Here, `Z^+` denotes the set of all non-negative integers.

Determine whether or not the definition of * On Z^+ , defined * by a*b=a b gives a binary operation. In the event that * is not a binary operation give justification of this. Here, Z^+ denotes the set of all non-negative integers.

Determine whether or not the definition of * On R , define * by a*b=a+4b^2 gives a binary operation. In the event that * is not a binary operation give justification of this. Here, R denotes the set of all real numbers.

Determine whether or not the definition of * On Z^+ , defined * by a * b=a-b gives a binary operation. In the event that * is not a binary operation give justification of this. Here, Z^+ denotes the set of all non-negative integers.

Determine whether or not the definition of * On Z^+ , define * by a*b=|a-b| gives a binary operation. In the event that * is not a binary operation give justification of this. Here, Z^+ denotes the set of all non-negative integers.

Determine whether or not the definition of * On Z^+ , define * by a*b=a gives a binary operation. In the event that * is not a binary operation give justification of this. Here, Z^+ denotes the set of all non-negative integers.

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this. (i) On Z^+ , define ∗ by a ∗ b = a – b (ii) On Z^+ , define ∗ by a ∗ b = ab (iii) On R , define ∗ by a ∗ b = ab^2 (iv) On Z^+ , define ∗ by a ∗ b = |a – b| (v) On Z^+ , define ∗ by a ∗ b = a

Is * defined by a * b=(a+b)/2 is binary operation on Z.

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:

The operation ** defined by a**b = b^(a) on the set Z = {0,1,2,3,. . . } . Prove that ** is not a binary operation on Z.