Let A and B be two sets having 5 and 7 elements respectively.write the minimum and maximum number of elemts in
(ii) `A cap B`

Answer

To solve the problem of finding the minimum and maximum number of elements in the intersection of two sets A and B, we can follow these steps:

### Step 1: Understand the Sets
Let:
- Set A has 5 elements.
- Set B has 7 elements.

### Step 2: Define Intersection
The intersection of two sets, denoted as \( A \cap B \), consists of elements that are common to both sets.

### Step 3: Determine the Minimum Number of Elements in \( A \cap B \)
The minimum number of elements in the intersection occurs when there are no common elements between the two sets. This means:
- If all elements in A are different from all elements in B, then:
\[ \text{Minimum } |A \cap B| = 0 \]

### Step 4: Determine the Maximum Number of Elements in \( A \cap B \)
The maximum number of elements in the intersection occurs when all elements of the smaller set (in this case, set A) are also in the larger set (set B). Since set A has 5 elements, the maximum number of common elements can be:
- All 5 elements of A can be common with B, provided that those elements are also in B. Therefore:
\[ \text{Maximum } |A \cap B| = 5 \]

### Conclusion
Thus, the minimum and maximum number of elements in the intersection \( A \cap B \) are:
- Minimum: \( 0 \)
- Maximum: \( 5 \)

### Summary
- Minimum number of elements in \( A \cap B = 0 \)
- Maximum number of elements in \( A \cap B = 5 \)

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