For each binary operation * defined below, determine whether * is commutative or associative.(i) `O n Z , d efin e a*b = a - b`(ii) `O n Q , d efin e a*b = a b + 1`(iii) `O n Q , d efin e a*b =(a b)/2`(iv) `O

For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a ∗ b = a – b (ii) On Q , define a ∗ b = ab + 1 (iii) On Q , define a ∗ b = (ab)/(2) (iv) On Z^+ , define a ∗ b = 2^(ab) (v) On Z^+ , define a ∗ b = a^(b) (vi) On R – {– 1} , define a ∗ b = (a)/(b+1)

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (ii) On Q, define '**' by a**b=ab-1

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (vi) On Z^(+) , define '**' by a**b=a^(b)

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (v) On Z^(+) , define '**' by a**b=2^(ab)

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (iv) On Q, define '**' by a**b=(ab)/(3)

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (iii) On Q, define '**' by a**b=(ab)/(4)

For each binary operation '**' defined below, determine whether '**' is commutative and whether '**' is associative : (vii) On R-{-1} , define '**' by a**b=(a)/(b+1)

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