We are asked to find $n(A \cup B)$, which represents the number of elements in the union of sets $A$ and $B$. The union of two sets contains all the distinct elements that are in either set.

Step-by-step solution:

Given:

• $A = \{ h, o, u, s, t, n \}$
• $B = \{ c, o, u, g, a, r, s \}$

The union $A \cup B$ contains all the distinct elements from both sets $A$ and $B$.

First, list all the elements from both sets:

• Elements of $A$: $\{ h, o, u, s, t, n \}$
• Elements of $B$: $\{ c, o, u, g, a, r, s \}$

Now, combine the elements of $A$ and $B$, keeping only distinct elements (i.e., removing duplicates): $A \cup B = \{ h, o, u, s, t, n, c, g, a, r \}$

Distinct elements in $A \cup B$:

• $h, o, u, s, t, n, c, g, a, r$

There are 10 distinct elements in $A \cup B$.

Final answer:

$n(A \cup B) = 10$

Would you like further clarification or details on any part of this?

Here are 5 related questions to practice:

1. Find $n(A \cap B)$, the number of elements in the intersection of $A$ and $B$.
2. What is $n(A \cup D)$, where $D = \{ f, o, t, b, a, l \}$?
3. Find $n(U \setminus A)$, where $U$ is the universal set of all English alphabet letters.
4. How many elements are in $A \cup D \cup B$?
5. What is $n(A \cap D)$?

Tip: The union of two sets counts all elements from both sets, but duplicates are included only once!