Given the sets $A$ and $B$:

$A = \{13, 22, 31, 40\}$ $B = \{14, 22, 41\}$

We need to find the number of elements in the union of sets $A$ and $B$, denoted as $A \cup B$.

Union of Sets $A$ and $B$

The union of two sets contains all unique elements from both sets. Let's list all the elements from $A$ and $B$:

• From $A$: $13, 22, 31, 40$
• From $B$: $14, 22, 41$

Now, combine these lists and remove any duplicates:

$A \cup B = \{13, 22, 31, 40, 14, 41\}$

Counting the Elements

Count the number of unique elements in $A \cup B$:

• $13$
• $22$
• $31$
• $40$
• $14$
• $41$

There are 6 unique elements.

Thus, the value of $n(A \cup B)$ is:

$n(A \cup B) = 6$

Would you like details or have any questions about this problem? Here are some related questions for further practice:

1. Find $n(A \cap B)$ for the sets defined above.
2. Determine $n(A \setminus B)$.
3. Determine $n(B \setminus A)$.
4. If $C = \{ x : x \text{ is a two-digit number such that the difference of its digits is 4}$, find $n(A \cup C)$.
5. If $D = \{ x : x \text{ is a two-digit number such that the sum of its digits is 6}$, find $n(D \cup B)$.
6. Verify if any element in $A \cap B$ exists.
7. Find the total number of two-digit numbers where the sum of the digits is 5.
8. Find the total number of two-digit numbers where the product of the digits is 6.

Tip: When dealing with sets and their unions and intersections, it's helpful to list out the elements clearly to avoid double-counting.