A relation on the set A = {x: |x| < 3, x ∈ Z}, where Z is the set of integers, is defined by R = {(x, y): y = |x|, x ≠ -2}. Then, the number of elements in the power set of R is 32, 16, 8, 64?

First consider R = {(x, y): y = |x|, x ≠ -2}
A = {-2, -1, 0, 1, 2} (since |x| < 3 and x ∈ Z) ⇒ R = {(0, 0), (1, 1), (2, 2), (-1, 1)} (since x ≠ -2)
There are 4 elements in R.
Power set contains all subsets of a given set. Hence the number of elements in the power set of R = 4C0 + 4C1 + 4C2 + 4C3 + 4C4 = 1 + 4 + 6 + 4 + 1 = 16.
Alternatively, using the result, No. of elements in power set = 2n, where 'n' is the total number of elements in a set, we get 24 = 16.