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

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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 $25$ students studied both mathematics and physics.


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 physics:

$N = S_1 \cap S_2$


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 $\card {S_1 \cup S_2}$ who studied either mathematics or physics is not more than $75$, the total number of students.

We have therefore:

\(\ds \card {S_1 \cup S_2}\) \(=\) \(\ds \card {S_1} + \card {S_2} - \card {S_1 \cap S_2}\) Cardinality of Set Union: 2 Sets
\(\ds \leadsto \ \ \) \(\ds 75\) \(\le\) \(\ds 40 + 60 - N\) where $N$ denotes the number of students who studied both mathematics and physics
\(\ds \leadsto \ \ \) \(\ds N\) \(\le\) \(\ds 25\) simplifying

$\blacksquare$


Sources

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