A & B are subsets of universal set U such that n(U) = 800, n(A) = 300, n(B) = 400 & n(A `cap` B) = 100. The number of elements in the set `A^(c ) cap B^(c )` is

Options

A. 100
B. 200
C. 300
D. 400

Answer

To find the number of elements in the set \( A^c \cap B^c \), we can use the relationship between the universal set, the sets \( A \) and \( B \), and their complements. Here’s the step-by-step solution:

### Step 1: Understand the Given Information
We are given:
- \( n(U) = 800 \) (the number of elements in the universal set \( U \))
- \( n(A) = 300 \) (the number of elements in set \( A \))
- \( n(B) = 400 \) (the number of elements in set \( B \))
- \( n(A \cap B) = 100 \) (the number of elements in the intersection of sets \( A \) and \( B \))

### Step 2: Calculate \( n(A \cup B) \)
Using the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the values:
\[ n(A \cup B) = 300 + 400 - 100 \]
\[ n(A \cup B) = 700 \]

### Step 3: Calculate \( n(A^c \cap B^c) \)
The complement of the union of two sets is given by:
\[ A^c \cap B^c = U - (A \cup B) \]
Thus, we can express the number of elements in \( A^c \cap B^c \) as:
\[ n(A^c \cap B^c) = n(U) - n(A \cup B) \]
Substituting the values we have:
\[ n(A^c \cap B^c) = 800 - 700 \]
\[ n(A^c \cap B^c) = 100 \]

### Final Answer
The number of elements in the set \( A^c \cap B^c \) is \( 100 \).

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