2. (40 points) Answer the following questions. (a) (10 points) Prove that D = {-e {0, 1)" I #0(x) is even ) is regular. (b) (10 points) Prove that E = { x E {0, 1)" | #1(x) is odd }is regular (c) E {0,1)' I #0(z) is even and #1(x) is odd } is regular by (20 points) Prove that F = { x constructing a product machine from D and E