If set A has 3 elements and set B has 6 elements, then the number of elements in `A uu B`?

Options

A. `6 le n (A cap B) le 8`
B. `6 le n(A cup B) le 9`
C. `0 le n(A cap B) le 2`
D. `1 le n(A cap B) le 3`

Answer

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

### Step 1: Identify the number of elements in each set
- Set A has 3 elements.
- Set B has 6 elements.

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

### Step 3: Determine the minimum number of elements in A ∪ B
- The minimum number of elements in A ∪ B occurs when all elements of set A are also in set B. In this case, the total number of distinct elements is equal to the number of elements in set B.
- Therefore, the minimum value of |A ∪ B| = |B| = 6.

### Step 4: Determine the maximum number of elements in A ∪ B
- The maximum number of elements in A ∪ B occurs when there are no common elements between sets A and B. In this case, we simply add the number of elements in both sets.
- Therefore, the maximum value of |A ∪ B| = |A| + |B| = 3 + 6 = 9.

### Step 5: Conclusion
- The number of elements in A ∪ B can range from 6 to 9, inclusive. Thus, we can express this as:
\[ 6 \leq |A ∪ B| \leq 9 \]

### Final Answer
The number of elements in A ∪ B is between 6 and 9.

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