Hi,
It is easier to write the complement of X and Y as X' and
Y' respectively.
We start with LHS, i.e.,
(X ∩ Y)' ∩ (X' U Y) ∩ (Y' U Y).
We deal with the first term and we want to remove the
complement by De Morgan's law, i.e.,
(X' U Y') ∩ (X' U Y) ∩ (Y' U Y).
Next, we observe that X' is common in the first two
terms, which can then be factored out via Distributive law,
i.e.,
[X' U (Y' ∩ Y)] ∩ (Y' U Y).
We proceed to simplify the second and third terms,
i.e.,
[X' U ∅] ∩ U.
Finally, we arrive at the desired result, i.e., X'.
Thanks.
Cheers,
Wen Shih