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:

\(\ds \card {S_1}\)

\(=\)

\(\ds 40\)

\(\ds \card {S_2}\)

\(=\)

\(\ds 60\)

\(\ds \card {S_3}\)

\(=\)

\(\ds 25\)

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:

\(\ds \card {S_2 \cup S_3}\)

\(=\)

\(\ds \card {S_2} + \card {S_3} - \card {S_2 \cap S_3}\)

Cardinality of Set Union: 2 Sets

\(\ds \leadsto \ \ \)

\(\ds 75\)

\(\le\)

\(\ds 60 + 25 - N\)

where $N$ denotes the number of students who studied both mathematics and physics

\(\ds \leadsto \ \ \)

\(\ds N\)

\(\le\)

\(\ds 10\)

simplifying

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $10 \ \text {(b)}$