Cardinality of Set Union/Examples

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Examples of Use of Cardinality of Set Union

Example: 3 Arbitrary Sets

Let $A_1, A_2, A_3$ be finite sets.

Let:

\(\ds \card {A_1}\) \(=\) \(\ds 10\)
\(\ds \card {A_2}\) \(=\) \(\ds 15\)
\(\ds \card {A_3}\) \(=\) \(\ds 20\)
\(\ds \card {A_1 \cap A_2}\) \(=\) \(\ds 8\)
\(\ds \card {A_2 \cap A_3}\) \(=\) \(\ds 9\)

Then:

$26 \le \card {A_1 \cup A_2 \cup A_3} \le 28$


Example: Examination Candidates

In a particular examination, there were $3$ questions.

All candidates attempted at least one of the questions.

$40$ candidates attempted question $1$.
$47$ candidates attempted question $2$.
$31$ candidates attempted question $3$.

Also, it was apparent that:

$9$ candidates attempted at least questions $1$ and $2$.
$15$ candidates attempted at least questions $1$ and $3$.
$11$ candidates attempted at least questions $2$ and $3$.

and:

exactly $6$ candidates attempted all $3$ questions.


It follows that $89$ candidates sat the examination in total.


Example: Student Subjects

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:


Mathematics and Physics

at least $25$ students studied both mathematics and physics.


Physics and Chemstry

at least $10$ students studied both physics and chemistry.


Mathematics and Chemistry

no more than $20$ students studied both mathematics and chemistry.