Question -
Answer -
(i) First we have toprove commutativity of *
Let a, b ∈ Z. then,
a * b = a + b – 4
= b + a – 4
= b * a
Therefore,
a * b = b * a, ∀ a, b ∈ Z
Thus, * is commutativeon Z.
Now we have to proveassociativity 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
Therefore,
a * (b * c) = (a * b)* c, for all a, b, c ∈ Z
Thus, * is associativeon Z.
(ii) Let e be the identityelement 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 theidentity element in Z with respect to *.
(iii) Let a ∈ Z 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
Thus, 8 – a is theinverse of a ∈ Z