4. The elements of sets $A, B$, and $C$ are some natural numbers that are greater than 20 and less than 30.

Furthermore, it holds that:

$A \cap B=\{23,28\}$,

$A \cap C=\{28,29\}$,

$B \cap C=\{28\}$,

$A \cup B=\{21,22,23,24,28,29\}$,

$A \cup C=\{21,22,23,25,26,27,28,29\}$, and

$B \cup C=\{n \in \mathbb{N}: 23 \leq n<30\}$.

Determine the sets $A, B$, and $C$.
4. The elements of sets $A, B$, and $C$ are some natural numbers that are greater than 20 and less than 30.

Given:

$$
\begin{aligned}
& A \cap B=\{23,28\} \\
& A \cap C=\{28,29\} \\
& B \cap C=\{28\} \\
& A \cup B=\{21,22,23,24,28,29\} \\
& A \cup C=\{21,22,23,25,26,27,28,29\} \text{ and} \\
& B \cup C=\{n \in \mathbb{N}: 23 \leq n<30\}
\end{aligned}
$$

Determine the sets $A, B$, and $C$.

## Solution.

The problem can be solved using Venn diagrams or another method. If another method is used, 2 points are awarded for a reasonable explanation, which are here associated with Venn diagrams.

Determine the elements of the set $B \cup C$.

$B \cup C=\{23,24,25,26,27,28,29\}$

1 POINT

From the given intersections $A \cap B=\{23,28\}, A \cap C=\{28,29\}, B \cap C=\{28\}$, it is clear that 28 is an element of all three intersections. We will plot the remaining elements of the intersections.

![](https://cdn.mathpix.com/cropped/2024_05_30_81794be492b317694280g-08.jpg?height=357&width=417&top_left_y=1301&top_left_x=268)

From the given unions of sets, we will plot the elements in the diagram.

![](https://cdn.mathpix.com/cropped/2024_05_30_81794be492b317694280g-08.jpg?height=362&width=417&top_left_y=1812&top_left_x=340)

$A=\{21,22,23,28,29\}$

1 POINT

$B=\{23,24,28\}$

$C=\{25,26,27,28,29\}$