If n(A union B)'=5,n(A without B)=3 and n(B)=7,then find n(u)

We are given the following information:

- \( n(A \cup B)' = 5 \), which means the number of elements in the complement of \( A \cup B \) is 5.
- \( n(A - B) = 3 \), which means the number of elements in \( A \) but not in \( B \) is 3.
- \( n(B) = 7 \), which means the number of elements in \( B \) is 7.

We need to find \( n(U) \), the number of elements in the universal set \( U \).

### Step-by-step solution:

1. From the complement, we know:
   \[
   n(A \cup B)' = n(U) - n(A \cup B)
   \]
   So,
   \[
   5 = n(U) - n(A \cup B)
   \]
   This gives:
   \[
   n(A \cup B) = n(U) - 5
   \]

2. Next, use the formula for the union of two sets:
   \[
   n(A \cup B) = n(A) + n(B) - n(A \cap B)
   \]

3. We know \( n(A - B) = 3 \), which represents the part of \( A \) not in \( B \). Therefore, the total number of elements in \( A \) can be written as:
   \[
   n(A) = n(A - B) + n(A \cap B)
   \]
   So,
   \[
   n(A) = 3 + n(A \cap B)
   \]

4. Substitute this into the formula for \( n(A \cup B) \):
   \[
   n(A \cup B) = (3 + n(A \cap B)) + 7 - n(A \cap B)
   \]
   Simplifying:
   \[
   n(A \cup B) = 3 + 7 = 10
   \]

5. Now, we know that \( n(A \cup B) = n(U) - 5 \), so:
   \[
   10 = n(U) - 5
   \]
   Solving for \( n(U) \):
   \[
   n(U) = 15
   \]

Thus, the number of elements in the universal set \( n(U) = 15 \).

---

Would you like more details or have any questions about the solution?

Here are 5 related questions to deepen your understanding:
1. How do you calculate \( n(A \cap B) \) if more data was provided?
2. What is the role of complements in set theory problems like this?
3. How can you extend this problem if \( n(A \cup B) \) were directly given?
4. How does \( n(A - B) \) differ from \( n(A \cup B) \) in interpretation?
5. How would the answer change if the value of \( n(B) \) was different?

**Tip:** When solving set problems, always sketch a Venn diagram—it helps visualize the relationships between the sets.

If n(A union B)' = 5, n(A without B) = 3 and n(B) = 7, then find n(U)

Learn how to find the universal set n(U) given n(A union B)' = 5, n(A without B) = 3, and n(B) = 7. This problem involves core set theory concepts such as union, complement, and intersection. Suitable for students in Grades 10-12.

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