Cardinality of Set Union/Examples/Student Subjects/Physics and Chemstry

Example of Use of Cardinality of Set Union
In a particular group of $75$ students, all studied at least one of the subjects mathematics, physics and chemistry.

All candidates attempted at least one of the questions.


 * $40$ students studied mathematics.
 * $60$ students studied physics.
 * $25$ students studied chemistry.

It follows that:
 * at least $10$ students studied both physics and chemistry.

Proof
Let:
 * $S_1$ denote the set of students who studied mathematics.
 * $S_2$ denote the set of students who studied physics.
 * $S_3$ denote the set of students who studied chemistry.

Knowledge of the total number of students gives us:
 * $S_1 \cup S_2 \cup S_3 = 75$

Let $N$ denote the number of students $N$ who studied both physics and chemistry:
 * $N = S_2 \cap S_3$

From the question:

The number of students $\card {S_2 \cup S_3}$ who studied either physics or chemistry is not more than $75$, the total number of students.

We have therefore: