Sets A and B have 5 and 7 elements respectively. What can be the minimum number of elements in `A uu B`?

Options

A. 5
B. 7
C. 12
D. 35

Answer

To find the minimum number of elements in the union of sets A and B (denoted as \( A \cup B \)), we can follow these steps:

### Step 1: Identify the number of elements in each set
- Set A has 5 elements.
- Set B has 7 elements.

### Step 2: Understand the concept of union
The union of two sets includes all the unique elements from both sets. If there are common elements between the two sets, they are only counted once in the union.

### Step 3: Maximize the overlap to minimize the union
To find the minimum number of elements in \( A \cup B \), we should maximize the overlap between the two sets. This means we want as many elements of set A to be included in set B.

### Step 4: Assume the elements of the sets
- Let’s assume the elements of set A are \( \{1, 2, 3, 4, 5\} \).
- To maximize overlap, we can assume that the first 5 elements of set B are the same as those in set A. Therefore, set B can be \( \{1, 2, 3, 4, 5, 6, 7\} \).

### Step 5: Calculate the union
Now, when we take the union of sets A and B:
- \( A = \{1, 2, 3, 4, 5\} \)
- \( B = \{1, 2, 3, 4, 5, 6, 7\} \)

The union \( A \cup B \) will be:
- \( A \cup B = \{1, 2, 3, 4, 5, 6, 7\} \)

### Step 6: Count the unique elements
The total number of unique elements in \( A \cup B \) is 7.

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

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