If `n(U)= 70, n(A) = 20, n(B)= 30 and n(A nn B) =10`. Then, `n(A^(c ) uu B^(c ))` is equal to

Options

A. 40
B. 50
C. 60
D. 30

Answer

To solve the problem step by step, we will use the principles of set theory along with the given values.

### Given:
- \( n(U) = 70 \) (Total number of elements in the universal set)
- \( n(A) = 20 \) (Number of elements in set A)
- \( n(B) = 30 \) (Number of elements in set B)
- \( n(A \cap B) = 10 \) (Number of elements in the intersection of sets A and B)

### We need to find:
- \( n(A^c \cup B^c) \) (Number of elements in the union of the complements of sets A and B)

### Step 1: Use the formula for the union of complements
We can use the formula:
\[ n(A^c \cup B^c) = n(U) - n(A \cap B) \]
This formula is derived from De Morgan's laws, which state that the complement of the intersection of two sets is equal to the union of their complements.

### Step 2: Calculate \( n(A \cap B) \)
We already have \( n(A \cap B) = 10 \).

### Step 3: Substitute the values into the formula
Now, we can substitute the known values into the formula:
\[ n(A^c \cup B^c) = n(U) - n(A \cap B) \]
\[ n(A^c \cup B^c) = 70 - 10 \]

### Step 4: Perform the calculation
\[ n(A^c \cup B^c) = 60 \]

### Final Answer:
Thus, \( n(A^c \cup B^c) = 60 \).

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