Help me to prove this: If X ∪ Y = X ∪ Z and X' ∪ Y = X' ∪ Z, then Y = Z

Prove: If X ∪ Y = X ∪ Z and X' ∪ Y = X' ∪ Z, then Y = Z ?

Since X U Y = X U Z and X' U Y = X' U Z, we have
(X U Y) ∩ (X' U Y) = (X U Z) ∩ (X' U Z).

Rewrite this with the Distributive Law:
(X ∩ X') U Y = (X ∩ X') U Z

This implies that
∅ U Y = ∅ U Z ==> Y = Z.

I hope this helps!

Recall if A = B and C = D then A ∩ C = B ∩ D. So if A = X U Y, B = X U Z, C = X' U Y, and D = X' U Z, then A ∩ C = (X U Y) ∩ (X' U Y) = Y U (X ∩ X') = Y U ∅ = Y. Similarly B ∩ D = (X U Z) ∩ (X' U Z) = Z U (X ∩ X') = Z U ∅ = Z. But A ∩ C = B ∩ D, hence Y = Z.