How to solve this problem using set theory?

Question

$36$ students took English and Math test. $25$ passed English, and $28$ passed Math. $20$ passed both subjects.

a. How many students failed both subject?
b. How many students passed english only?
c. How many students passed math only?

Answer 1

If $E$ is the set of the students that passed English and $M$ are those that passe Math then you have:

$|E|=25$, $|M|=28$, $|E\cap M|=20$

From the formula

$$|E\cup M|=|E|+|M|-|E\cap M|$$

you get that $|E\cup M|=33$, which means that 33 student passed at least one subject.

I guess you can go on from here...

If you prefer diagrams instead of formulas, you can try to draw something like this: http://mrsgsmathclass.com/math%20pics/Probability/Venn3.jpg

http://mrsgsmathclass.com/Probability%20Pages/Venn%20Diagrams%20and%20Counting.html

Perhaps I should have also added this link: http://en.wikipedia.org/wiki/Inclusion-exclusion_principle

Answer 2

Denote by $S$ the set of all students, by $E$ the set of all students that passed English and by $M$ the set of all students that passed Math. Then what you know is: $|S|=36$, $|E|=25$, $|M|=28$ and $|E\cap M|=20$. Then what you are looking for is:
a. $|(S\setminus E)\cap (S\setminus M)|=$
b. $|E\setminus M|=$
c. $|M\setminus E|=$
So you can use De Morgan's laws and Martin Sleziak's answer from here on to solve...