Cardinality of Set Union/Examples/Student Subjects/Mathematics and Chemistry
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.
Also:
- exactly $5$ students studied all $3$ subjects.
It follows that:
- no more than $20$ students studied both mathematics 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 mathematics and chemistry:
- $N = S_1 \cap S_3$
From the question:
\(\ds \card {S_1}\) | \(=\) | \(\ds 40\) | ||||||||||||
\(\ds \card {S_2}\) | \(=\) | \(\ds 60\) | ||||||||||||
\(\ds \card {S_3}\) | \(=\) | \(\ds 25\) |
First we calculate how many students took just mathematics or chemistry, but who did not take physics.
We have
\(\ds \card {\paren {S_1 \cup S_3} \setminus S_2}\) | \(=\) | \(\ds \card {\paren {S_1 \cup S_2 \cup S_3} \setminus S_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \card {S_1 \cup S_2 \cup S_3} - \card {S_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 75 - 60\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15\) |
So only $15$ students did not take physics.
Thus no more than $15$ students can have taken both mathematics and chemistry, without taking physics.
However, we are also told that $5$ students took all $3$ courses.
So, in addition to the maximum $15$ who took mathematics and chemistry, without taking physics, this takes the total to a maximum of $20$ students all told who took both mathematics and chemistry.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $10 \ \text {(c)}$