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

Listing Method; or
Membership Table.

Given
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
X = {2, 5, 8, 9}
Y = {1, 4, 6, 7, 8}
Z = {5, 7, 9}

Respuesta :

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

[tex]X\cap(Y\cup Z)=(X\cap Y)\cup (X\cap Z)[/tex]

Let's compute [tex]X\cap (Y\cup Z)[/tex] first. We have

[tex](Y\cup Z)=\{1, 4, 5, 6, 7, 8, 9\}[/tex]

If we intersect this set with X, we have

[tex]X\cap (Y\cup Z) = \{5, 8, 9\}[/tex]

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

[tex]X\cap Y = \{8\},\quad X\cap Z = \{5,9\}[/tex]

And their union is again [tex]\{5,8,9\}[/tex]

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