(i) Let us prove the commutativity of *
Let a, b ∈ Z. Then,
a * b = a + b – 4
= b + a – 4
= b * a
So,
a * b = b * a, ∀ a, b ∈ Z
Thus, * is commutative on Z.
Now, let us prove associativity of Z.
Let a, b, c ∈ Z. Then,
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
So,
a * (b * c) = (a * b) * c, for all a, b, c ∈ Z
Thus, * is associative on Z.
(ii) Let e be the identity element in Z with respect to *
Such that, 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
Thus, 4 is the identity element in Z with respect to *.
(iii) Let a and b ∈ Z be the inverse of a. Then,
a * b = e = b * a
a * b = e and b * a = e
a + b – 4 = 4 and b + a – 4 = 4
b = 8 – a ∈ Z
So, 8 – a is the inverse of a ∈ Z