Suppose that:
U = {x x is a letter in the English alphabet}
A = {h, o, u, s, t, n}
B = {c, o, u, g, a, r, s}
D = { f , o, t, b, a, l}
Find each of the following values. . n( A∪ B)

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!

Suppose that: U = {x | x is a letter in the English alphabet}, A = {h, o, u, s, t, n}, B = {c, o, u, g, a, r, s}, D = {f, o, t, b, a, l}. Find each of the following values: n(A ∪ B).

Learn how to find the union of sets A and B, including the cardinality (n(A ∪ B)) of their distinct elements. This example uses basic set theory concepts, perfect for students in Grades 6-8.

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