First of all you must be sure to distinguish between the U which
means the universal set, and the symbol ∪ for union.
Let U={a,b,c,d,e,f,g,h}
A={a,c,d}
B={b,c,d}
C={b,e,f,g,h}
Find the set: (C'∩A)U(C'∩B')
[Notice you used U instead of ∪]
(C' ∩ A) ∪ (C' ∩ B')
Replace the letters by the sets:
({b,e,f,g,h}' ∩ {a,c,d}) ∪ ({b,e,f,g,h}' ∩ {b,c,d}')
The apostrophes represent the complement sets which
means to take all the elements of the universal set
EXCEPT those elements in the set before the apostrophe.
So the above becomes:
({a,c,d} ∩ {a,c,d}) ∪ ({a,c,d} ∩ {a,e,f,g,h})
Inside the first parentheses, the ∩ means to include
just the elements that are common to both sets on each
side of the ∩, so since all the elements of {a,c,d}
are in common with the elements of {a,c,d}, because it's
the same set, the first parentheses just becomes {a,c,d}
So we have:
{a,c,d} ∪ ({a,c,d} ∩ {a,e,f,g,h})
In the remaining parentheses, the ∩ means to include
just the elements that are common to both sets on each
side of the ∩, and since "a" is the only element common
to both {a,c,d} and {a,e,f,g,h} the parentheses becomes {a},
and we have:
{a,c,d} ∪ {a}
The ∪ means to use all the elements in both sets regardless
of whether they are in common or not, but there is no need
to list any element twice, so the final answer is
{a,c,d}
Edwin