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united nepal publication (p) Ltd  10 class united speedy maths 10 
Union, overlapping, Disjoint sets 

Let's solve the given problems step by step.

### Question 1(a)
**Given:**
- \( n(U) = 250 \)
- \( n(P) = 100 \)
- \( n(Q) = 75 \)
- \( n(P \cap Q) = 25 \)

We need to show that \( n(P \cup Q) = 100 \) and illustrate the information in a Venn diagram.

**Solution:**

Using the formula for the union of two sets:
\[ n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \]

Substitute the given values:
\[ n(P \cup Q) = 100 + 75 - 25 = 150 \]

This seems to contradict the problem statement which asks to show \( n(P \cup Q) = 100 \). There might be an error in the problem statement or a misinterpretation. The usual formula shows it to be 150.

### Question 1(b)
**Given:**
- \( n(S) = 80\% \) of total students
- \( n(M) = 90\% \) of total students

We need to show:
1. \( n(S \cap M) = 70\% \) of total students
2. \( n(M) = 20\% \) of total students

**Solution:**

If we assume there are 100 students in total:
- \( n(S) = 80 \)
- \( n(M) = 90 \)
- \( n(S \cap M) = 70 \)

We need to show \( n(M) = 20 \) which seems to be misinterpreted. It should rather be something related to \( n(S \cup M) \).

### Question 1(c)
**Given:**
- \( n(U) = 112 \)
- \( n(C) = 56 \)
- \( n(D) = 45 \)
- \( n(\overline{C} \cup \overline{D}) = 0 \)
- \( n(C \cap D) = 11 \)
- \( n(C) = 67 \)

**Solution:**

For \( n(\overline{C} \cup \overline{D}) = 0 \), it implies all elements are either in \( C \) or \( D \).

We use the principle of inclusion and exclusion:
\[ n(C \cup D) = n(C) + n(D) - n(C \cap D) \]
\[ n(C \cup D) = 56 + 45 - 11 = 90 \]

Since \( n(U) = 112 \) and all elements are in \( C \cup D \), this seems inconsistent with the total number of elements.

### Question 2(a)
**Given:**
- \( n(U) = 60 \)
- \( n(A) = 40 \)
- \( n(B) = 15 \)
- \( n(A \cup B) = 15 \)

**Solution:**

Using the principle of inclusion and exclusion:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Given \( n(A \cup B) = 15 \):
\[ 15 = 40 + 15 - n(A \cap B) \]
\[ n(A \cap B) = 40 \]

This implies there might be an error because \( n(A \cap B) \) cannot exceed \( n(B) \).

Let's focus on confirming this Venn diagram and the calculations.

### Question 2(b)
**Given:**
- \( n(U) = 56 \)
- \( n(P) = 32 \)
- \( n(Q) = 20 \)
- \( n(P \cap Q) = 12 \)

**Solution:**

Using the formula for the union of two sets:
\[ n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \]
\[ n(P \cup Q) = 32 + 20 - 12 = 40 \]

Thus, \( n(P \cup Q) = 40 \).

I suggest verifying the interpretations and calculations as some provided data and questions seem to have inconsistencies.

Would you like detailed steps for any specific question or a different approach to solving any of these problems?

Here are 5 follow-up questions you might want to ask:

1. Can you provide more details on how to draw the Venn diagram for question 1(a)?
2. Could you explain the principle of inclusion and exclusion in more detail?
3. How can we address inconsistencies in provided problem statements?
4. Could you solve another example involving Venn diagrams and set operations?
5. What are the common mistakes to avoid when working with Venn diagrams in set theory?

**Tip:** Always double-check problem statements and given data for consistency before proceeding with calculations.

Learn how to solve set theory problems using Venn diagrams with detailed step-by-step solutions. This problem covers topics such as union, intersection, and disjoint sets, illustrating the principle of inclusion and exclusion. Suitable for high school students.

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