# Cardinality of Set Union/Examples/Examination Candidates

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## Example of Use of Cardinality of Set Union

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.

## Proof

Let:

- $S_1$ denote the set of candidates who attempted question $1$.
- $S_2$ denote the set of candidates who attempted question $2$.
- $S_3$ denote the set of candidates who attempted question $3$.

The number of candidates $N$ who sat the examination in total is therefore:

- $N = S_1 \cup S_2 \cup S_3$

From Cardinality of Set Union: 3 Sets:

\(\ds \card {S_1 \cup S_2 \cup S_3}\) | \(=\) | \(\ds \card {S_1} + \card {S_2} + \card {S_3}\) | ||||||||||||

\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}\) | |||||||||||

\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \card {S_1 \cap S_2 \cap S_3}\) |

From the question:

\(\ds \card {S_1}\) | \(=\) | \(\ds 40\) | ||||||||||||

\(\ds \card {S_2}\) | \(=\) | \(\ds 47\) | ||||||||||||

\(\ds \card {S_3}\) | \(=\) | \(\ds 31\) | ||||||||||||

\(\ds \card {S_1 \cap S_2}\) | \(=\) | \(\ds 9\) | ||||||||||||

\(\ds \card {S_1 \cap S_3}\) | \(=\) | \(\ds 15\) | ||||||||||||

\(\ds \card {S_2 \cap S_3}\) | \(=\) | \(\ds 11\) | ||||||||||||

\(\ds \card {S_1 \cap S_2 \cap S_3}\) | \(=\) | \(\ds 6\) |

Hence:

\(\ds \card {S_1 \cup S_2 \cup S_3}\) | \(=\) | \(\ds 40 + 47 + 31\) | ||||||||||||

\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 9 - 15 - 11\) | |||||||||||

\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 6\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 89\) |

$\blacksquare$

## Sources

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $9$