Set A has 4 elements and set Bhas 7 elements. What can be the minimum number of elements in `A uu B`?

Options

A. 6
B. 7
C. 8
D. 9

Answer

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

### Step 1: Identify the number of elements in each set
- Set A has 4 elements.
- Set B has 7 elements.

### Step 2: Understand the concept of union
The union of two sets A and B, denoted as \( A \cup B \), includes all the elements that are in A, in B, or in both.

### Step 3: Determine the scenario for minimum elements in the union
To find the minimum number of elements in \( A \cup B \), we need to consider the maximum overlap between the two sets. This means we want to maximize the number of elements that are common to both sets.

### Step 4: Calculate the maximum overlap
- If all 4 elements of set A are also present in set B, then the overlap is 4 elements.
- Set B has 7 elements, and if 4 of those are already in set A, then the remaining elements in set B will be \( 7 - 4 = 3 \).

### Step 5: Calculate the total number of unique elements in the union
Using the formula for union:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Where:
- \( |A| = 4 \) (number of elements in A)
- \( |B| = 7 \) (number of elements in B)
- \( |A \cap B| = 4 \) (maximum overlap)

Substituting the values:
\[ |A \cup B| = 4 + 7 - 4 = 7 \]

### Conclusion
Thus, the minimum number of elements in \( A \cup B \) is **7**.