How do i solve this? Please show work

Question

Given that E and F are sets and n(E \ F) =12 and n(F \ E) = 30, what is the smallest possible value for n(E ∪ F)?

Answer 1

Given sets E and F.

Suppose n(E∩F) = m

 

The number of elements in E are the number of elements in E that are not in F, n(E\F), and also the number of elements that are in both E and F, n(E∩F).  ∴ n(E) = n(E\F) + n(E∩F) = n(E\F) + m

 

The number of elements in F are the number of elements in F that are not inE, n(F\E), and also the number of elements that are in both F and E, n(F∩E). ∴ n(F) = n(F\E) + n(E∩F)  = n(F\E) +m

 

Now n(E∪F) = n(E) + n(F) - n(E∩F) = ( n(E\F) + m) + (n(F\E) +m) - m = n(E\F) + (n(F\E) + m

From this the union is minimum when m = 0, i.e. the sets are disjoint => E∩F= Ø

 

Since n(E/F) = 12 and n(F/E) = 30, the minimum value of n(E∪F) = n(E\F) + (n(F\E) + m = 12 + 30 + 0 =42