Question

Prove that using: X ∩ (Y ∪ Z ) =(X∩Y) ∪ (Y ∩ Z )

Answer 1

There should be a typo in the question: the correct formula would be

$X\cap(Y\cup Z)=(X\cap Y)\cup (X\cap Z)$

Let's compute $X\cap (Y\cup Z)$ first. We have

$(Y\cup Z)=\{1, 4, 5, 6, 7, 8, 9\}$

If we intersect this set with X, we have

$X\cap (Y\cup Z) = \{5, 8, 9\}$

So, that would be the left hand side. For the right hand side, we have

$X\cap Y = \{8\},\quad X\cap Z = \{5,9\}$

And their union is again $\{5,8,9\}$