If A and B are two sets, then n(A)+ n(B) is equal to

Options

A. `n(A uu B)`
B. `n(A nn B)`
C. `n(A uu B) -n(A nn B)`
D. `n(A uu B) + n( A nn B)`

Answer

To find the value of \( n(A) + n(B) \) where \( A \) and \( B \) are two sets, we can use the concept of set theory and Venn diagrams. Here’s a step-by-step solution:

### Step 1: Understand the Definitions
- \( n(A) \) represents the number of elements in set \( A \).
- \( n(B) \) represents the number of elements in set \( B \).
- \( n(A \cap B) \) represents the number of elements that are common to both sets \( A \) and \( B \).
- \( n(A \cup B) \) represents the number of elements in either set \( A \), set \( B \), or both.

### Step 2: Use the Formula for Union of Sets
The relationship between the sizes of the sets can be expressed with the formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
This formula accounts for the fact that elements in the intersection \( A \cap B \) are counted twice when we simply add \( n(A) \) and \( n(B) \).

### Step 3: Rearranging the Formula
To find \( n(A) + n(B) \), we can rearrange the formula:
\[ n(A) + n(B) = n(A \cup B) + n(A \cap B) \]
This shows that the sum of the sizes of the two sets is equal to the size of their union plus the size of their intersection.

### Step 4: Conclusion
Thus, the answer to the question "If A and B are two sets, then \( n(A) + n(B) \) is equal to" can be summarized as:
\[ n(A) + n(B) = n(A \cup B) + n(A \cap B) \]