Proof, set of solutions?

Question

Mathematics

Let three sets of solutions of some equations be given
A: x = pi*n, n - Z
B: x = pi/2 + pi*n, n - Z
C: x = pi/2*n, n - Z
Prove that AUB=C!
I don't know how to prove it, please tell me

Answer 1

it suffices to show that any solution from C is either a solution from A or a solution from B.
For even n=2k we get x = pi/2*(2k) = pi*k, k - Z, belongs to the set A
For odd n = (2k+1) we get x = pi/2*(2k+1) = pi*k + pi/2, k - Z, belongs to the set B. (
Sufficiency) is proved
that solutions from A and B belong to C, but
A is simpler there: x = pi*n = pi/2 * (2n) = pi/2 * k, k - Z
B:x = pi/2 + pi*n = pi /2 + pi/2*(2n) = pi/2 * (2n + 1) = pi/2 * k, k -
Z