# How to find the intersection part in a Venn diagram?

My problem: There are $$20$$ students in a class. $$13$$ of them study chemistry and $$16$$ of them study physics. $$3$$ of them do neither.

My workings:

$$13 + 16 = 29$$, but there are only $$20$$ people in the class, that means some people have to do both right? But how do I determine $$C \cap P$$? So, I said $$20 -3 =17$$, then $$C \cup P$$ must have a total of $$17$$ people. But then I am stuck.

Can someone please visually represent this? Mathematically showing how to find $$C \cap P$$ is also fine.

As you correctly observed, $$17$$ study either chemistry or physics. As $$16$$ study physics only $$1$$ student studies only chemistry. Similarly, $$4$$ students study only physics. This results in $$12$$ students studying both.

You have a big set of Students $$S$$ and you have two subsets $$A,B$$ of students who study chemistry or physics respectively. You know that $$\#(A\cup B)=20-3=17$$. Also note that $$A\cap B = (A^c\cup B^c)^c$$ where $$c$$ denotes the complement. But you do know how many people do NOT study chemistry or physics.

Do you know how to continue from here?

If $$C$$ is the set containing all chemistry students and $$P$$ is the set containing all physics students, we have:

$$|C| + |P| - |C \cap P| = 20 - 3 \iff 13 + 16 - |C \cap P| = 17$$ $$\iff |C \cap P| = 29 - 17 = 12$$

We thus find $$12$$ students studying both topics, $$13 - 12 = 1$$ studying only chemistry and $$16 - 12 = 4$$ studying only physics. The Venn diagram looks as follows: 