SOLUTION: Draw a Venn diagram and use the given information to fill in the number of elements in each region. ​n(A) = 24​, ​n(B)=16​, ​n(A ∪ ​B)=

Draw two intersecting circles, A and B.


The circle A symbolizes (represents) the finite set A containing 24 elements.

The circle B symbolizes (represents) the finite set B containing 16 elements.


The common part of the circles A and B represents the intersection of these two sets.


By knowing that the union  (A U B)  contains 31 element, we can calculate the number of elements in the intersection (A n B).


It is  n(A n B) =  n(A) + n(B) - n(A U B) =         (1)

                =  24 + 16 - 31 = 40 - 31 = 9. 


Why ?  - It is easier to me to prove an equivalent formula

n(A U B) = n(A) + n(B) - n(A n B).                  (2)


    Indeed, when we count  n(A U B)  as the sum  n(A) + n(B),  we count elements of the intersection  TWICE. 

    Therefore, we need to subtract  n(A n B) from the sum  n(A) + n(B)  to get the exact number of elements in  n(A U B).


So, the formula (2) is proved, and, with it,  the formula (1) and the following calculations are justified.


OK.  Now we know the number of elements in the intersection (although the problem did not ask us about it. So, it is to wider your horizon).


Now to the problem question: find  n(A').


n(A') = n(A U B) - n(A) = 31 - 24 = 7.


It is so obvious that you can get it on your own if you will think 7 seconds about it looking in your plot.