Finding the maximum percentage of people who are not in A or B

Question

60% of the population is in A 50% of the population is in B

To get the maximum number of people in neither, is it right to find the maximum number of people in both (50% in this case) and plug that in to the inclusion-exclusion formula?

(A or B) = A + B - (A and B) + None

So would the max of neither be 40%?

Answer 1

Another way to look at it: $$\begin{align}|A\cup B|+none&=100\% \Rightarrow \\ none&=100\%-|A\cup B|=100\%-(|A|+|B|-|A\cap B|)=\\ &=100\%-60\%-50\%+|A\cap B|=\\ &=|A\cap B|-10\%.\end{align}$$ Hence, none (neither) is maximum $40\%$ when $|A\cap B|$ is maximum $50\%$.

Answer 2

Your equation is wrong, but your result is right. We have

$$ |A\cup B|=|A|+|B|-|A\cap B| $$

(which corresponds to your equation without the “None” term), and since you want to minimize $|A\cup B|$ with $|A|$ and $|B|$ given, you need to maximize $|A\cap B|$.