As I was doing homework I noticed the answer in the back of the book read as line 1 in the picture above. If I were to put the answer in line 2 would that answer be wrong?
Does the bar there mean topological closure? If so, then the answer is yes, they're equal. If the bar means set complement, then see Plato's response above.
I'll write the closure of a set X as as cl(X).
Suppose N is a closed set containing (A union B union C). Then N is a closed set containing A, so cl(A) contained in N. Likewise, cl(B) and cl(C) are contained in N. Thus (cl(A) union cl(B) union cl(C)) is contained in N. Since N was any closed set containing A, B, and C, have that (cl(A) union cl(B) union cl(C)) is contained in the intersection of all such closed sets. Thus (cl(A) union cl(B) union cl(C)) is a subset of cl(A union B union C).
Now notice that cl(a) is a closed set containing A, cl(B) is a closed set containing B, and cl(C) is a closed containing C. Thus (cl(A) union cl(B) union cl(C)) is a closed set that contains (A union B union C). Thus (cl(A) union cl(B) union cl(C)) contains cl(A union B union C).
Putting those two observations together proves that (cl(A) union cl(B) union cl(C)) = cl(A union B union C).
This obviously generalizes to any finite number of sets.