If the cardinality of a set A is 4 and that of a set B is 3, then what is the cardinality of the set `A Delta B`?

Options

A. 1
B. 5
C. 7
D. Cannot be determined as the sets A and B are not given

Explanation

To find the cardinality of the set \( A \Delta B \) (the symmetric difference of sets A and B), we can follow these steps:

### Step 1: Understand the Definition of Symmetric Difference
The symmetric difference \( A \Delta B \) is defined as the set of elements that are in either set A or set B but not in their intersection. Mathematically, it can be expressed as:
\[ A \Delta B = (A \cup B) - (A \cap B) \]

### Step 2: Use the Cardinality Formula
The cardinality of the symmetric difference can be calculated using the formula:
\[ |A \Delta B| = |A| + |B| - 2|A \cap B| \]
where \( |A| \) is the cardinality of set A, \( |B| \) is the cardinality of set B, and \( |A \cap B| \) is the cardinality of the intersection of sets A and B.

### Step 3: Substitute the Known Values
From the problem, we know:
- \( |A| = 4 \)
- \( |B| = 3 \)

However, we do not have information about \( |A \cap B| \) (the number of elements common to both sets A and B). Therefore, we cannot determine the exact value of \( |A \Delta B| \) without knowing \( |A \cap B| \).

### Step 4: Conclusion
Since we do not have enough information to find \( |A \Delta B| \), we conclude that the cardinality of the set \( A \Delta B \) cannot be determined with the given information.

Thus, the answer is:
\[ \text{The cardinality of } A \Delta B \text{ cannot be determined.} \]

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