Answer :

(i) A is said to be closed on * if all the elements of a*b A. composition table is

(as i2 = -1)


As table contains all elements from set A, A is close for multiplication operation.


(ii) For associative, a× (b× c) = (a× b) ×c


1× (-i× i) = 1×1 = 1


(1× -i) ×i = -i× i = 1


a× (b× c) = (a× b) ×c, so A is associative for multiplication.


(iii) For commutative, a× b = b× a


1× -1 = -1


-1× 1 = -1


a× b = b× a, so A is commutative for multiplication.


(iv) For multiplicative identity element e, a× e = e× a = a where a A.


a× e = a


a(e-1) = 0


either a = 0 or e = 1 as a≠0 hence e = 1.


So, multiplicative identity element e = 1.


(v) For multiplicative inverse of every element of A, a*b = e where a, bA.


1×b1 = 1


b1 = 1


-1×b2 = 1


b2 = -1


i×b3 = 1



-i×b4 = 1



So, multiplicative inverse of A = {1, -1, -i, i}


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