Answers:

(i)

Consider,

a, b ∈ Z

a * b = a + b – 4

= b + a – 4

= b * a

a * b = b * a, ∀ a, b ∈ Z

Then, * is commutative on Z.

a * (b * c) = a * (b + c – 4)

= a + b + c -4 – 4

= a + b + c – 8

(a * b) * c = (a + b – 4) * c

= a + b – 4 + c – 4

= a + b + c – 8

a * (b * c) = (a * b) * c, for all a, b, c ∈ Z

Hence, * is associative on Z.

(ii)

Consider,

e be the identity element in Z with respect to *

a * e = a = e * a ∀ a ∈ Z

a * e = a and e * a = a, ∀ a ∈ Z

a + e – 4 = a and e + a – 4 = a, ∀ a ∈ Z

e = 4, ∀ a ∈ Z

Hence, 4 is the identity element in Z with respect to *.