Question -
Answer -
(i) First we have toprove commutativity of *
Let a, b тИИ Z. then,
a * b = a + b тАУ 4
= b + a тАУ 4
= b * a
Therefore,
a * b = b * a, тИА a, b тИИ Z
Thus, * is commutativeon Z.
Now we have to proveassociativity of Z.
Let a, b, c тИИ Z. then,
a * (b * c) = a * (b +c тАУ 4)
= a + b + c -4 тАУ 4
= a + b + c тАУ 8
(a * b) * c = (a + b тАУ4) * c
= a + b тАУ 4 + c тАУ 4
= a + b + c тАУ 8
Therefore,
a * (b * c) = (a * b)* c, for all a, b, c тИИ Z
Thus, * is associativeon Z.
(ii) Let e be the identityelement in Z with respect to * such that
a * e = a = e * a тИА a тИИ Z
a * e = a and e * a =a, тИА a тИИ Z
a + e тАУ 4 = a and e +a тАУ 4 = a, тИА a тИИ Z
e = 4, тИА a тИИ Z
Thus, 4 is theidentity element in Z with respect to *.
(iii) Let a тИИ Z and b тИИ Z be the inverse of a. Then,
a * b = e = b * a
a * b = e and b * a =e
a + b тАУ 4 = 4 and b +a тАУ 4 = 4
b = 8 тАУ a тИИ Z
Thus, 8 тАУ a is theinverse of a тИИ Z