Proof using Laws of Sets

Question

Using the laws of sets, prove:

a. $ A - (B ∩ C) = (A ∩ B^{c}) $ $∪ $ $(A∩C^{c})$

b. $ (A -B)^{c} $ × C $= (A^{c} $ × $C) $ $∪ $ $(B $ × C$)$

My answers:

a. $ A - (B ∩ C) $ =

$ A∩(B ∩ C)^{c} $

$ A∩(B^{c} ∪ C)^{c} $ (De Morgan's Law)

$ (A∩B^{c}) ∪ (A ∩C^{c})$ (Distribution Law)

b. $ (A -B)^{c} $ × C =

$ (A × C ) \cup ( B $ × $ C )$

Have I correctly solved question a?

I am stuck on question b, as I am unsure how to correctly apply the laws of sets.

Answer 1

Part $a$ is correct.

For part $b$, note that \begin{align} (A-B)^c = (A \cap B^c)=A^c \cup B \end{align}