Let P and Q be two sets such that `n(P uu Q) = 70 ,n (P )= 45 and n(Q) = 38, ` Draw a Venn diagram and find :
` n(P nn Q) `

Answer

To solve the problem step by step, we will use the information given about the sets P and Q.

### Step 1: Understand the given information
We are given:
- \( n(P \cup Q) = 70 \) (the number of elements in the union of sets P and Q)
- \( n(P) = 45 \) (the number of elements in set P)
- \( n(Q) = 38 \) (the number of elements in set Q)

### Step 2: Use the formula for the union of two sets
The formula for the number of elements in the union of two sets is:
\[ n(P \cup Q) = n(P) + n(Q) - n(P \cap Q) \]
where \( n(P \cap Q) \) is the number of elements in the intersection of sets P and Q.

### Step 3: Substitute the known values into the formula
Now, substituting the known values into the formula:
\[ 70 = 45 + 38 - n(P \cap Q) \]

### Step 4: Simplify the equation
Combine the numbers on the right side:
\[ 70 = 83 - n(P \cap Q) \]

### Step 5: Solve for \( n(P \cap Q) \)
Rearranging the equation to find \( n(P \cap Q) \):
\[ n(P \cap Q) = 83 - 70 \]
\[ n(P \cap Q) = 13 \]

### Step 6: Conclusion
Thus, the number of elements in the intersection of sets P and Q is:
\[ n(P \cap Q) = 13 \]

### Drawing the Venn Diagram
1. Draw two overlapping circles, one for set P and one for set Q.
2. Label the left circle as P and the right circle as Q.
3. In the overlapping area (the intersection), write the number 13.
4. In the part of circle P that does not overlap, write \( 45 - 13 = 32 \).
5. In the part of circle Q that does not overlap, write \( 38 - 13 = 25 \).

### Final Answer
- \( n(P \cap Q) = 13 \)

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