Let C={i,-1,-i,1}, where i is chosen so that i^2=-1.?

Question

The relation R on C given by xRy iff xy= 1or -1 is an equivalence relation on C. Give the partition of C associated with R.

Answer 1

Multiplying elements of C:

i*(-1) = - i,
i*(-i) = - i² = - (-1) = 1,
i*1 = i,
(-1)*(-i) = i,
(-1)*1 = - 1,
(-i)*1 = - i.

Thus, iR(-i) and 1R(-1). Hence, the partition of C given by R is

{ { 1, - 1 }, { i, -i } }.

Answer 2

It should be obvious that xRy <=> (x, y ∈ {1, -1} or x, y ∈{-i, i}) - if you multiply an i term with a 1 term you will get an i term.
Hence, the partition of C associated with R is {{1, -1}, {i, -i}}.