If A has 4 elements and B has 7 elements, then the minimum number of elements in A ∪ B is :

Options

A. 4
B. 7
C. 6
D. 5

Answer

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

### Step-by-Step Solution:

1. **Identify the Number of Elements in Each Set**:
- Set A has 4 elements.
- Set B has 7 elements.

2. **Understand the Concept of Union**:
- The union of two sets, denoted as \( A \cup B \), includes all the unique elements from both sets. If there are common elements, they are only counted once.

3. **Calculate the Maximum Possible Elements in the Union**:
- The maximum number of elements in \( A \cup B \) occurs when there are no overlapping elements. This is calculated as:
\[ |A \cup B|_{\text{max}} = |A| + |B| = 4 + 7 = 11 \]

4. **Determine the Minimum Possible Elements in the Union**:
- To find the minimum, we need to consider the maximum overlap between the two sets. The maximum number of overlapping elements can be limited by the smaller set, which in this case is set A with 4 elements.
- If all 4 elements of set A are also in set B, then the calculation for the minimum number of unique elements in the union is:
\[ |A \cup B|_{\text{min}} = |A| + |B| - \text{(number of overlapping elements)} \]
- If all 4 elements of A overlap with B, then:
\[ |A \cup B|_{\text{min}} = 4 + 7 - 4 = 7 \]

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

### Final Answer:
The minimum number of elements in \( A \cup B \) is **7**.