If `n(A) = 4 and n(B) =6`, then maximum number of elements in `A cup B` is :

Options

A. 2
B. 4
C. 6
D. 10

Answer

To find the maximum number of elements in the union of sets A and B, we can follow these steps:

### Step 1: Understand the given information
We are given:
- The number of elements in set A, denoted as \( n(A) = 4 \)
- The number of elements in set B, denoted as \( n(B) = 6 \)

### Step 2: Use the formula for the union of two sets
The formula for the number of elements in the union of two sets A and B is:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
where \( n(A \cap B) \) is the number of elements common to both sets A and B.

### Step 3: Determine the minimum number of elements in the intersection
To maximize \( n(A \cup B) \), we need to minimize \( n(A \cap B) \). The minimum value of \( n(A \cap B) \) is 0, which occurs when there are no elements common to both sets.

### Step 4: Substitute values into the formula
Now, substituting the values into the formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
\[ n(A \cup B) = 4 + 6 - 0 \]
\[ n(A \cup B) = 10 \]

### Conclusion
Thus, the maximum number of elements in \( A \cup B \) is 10.

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