Let A = (1, -1, i, -i) be the set of four 4th roots of unity. Prepare the composition table for multiplication on A and show that
(i) A is closed for multiplication,
(ii) multiplication is associative on A,
(iii) multiplication is commutative on A,
(iv) 1 is the multiplicative identity,
(v) every element in A has its multiplicative inverse.
(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, b∈A.
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}