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 AUB?

Options

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

Answer

To find the cardinality of the union of two sets A and B, we can use the following steps:

### Step 1: Understand the given information
We know that:
- The cardinality of set A, denoted as |A|, is 4.
- The cardinality of set B, denoted as |B|, is 3.

### Step 2: Define the union of two sets
The union of two sets A and B, denoted as A ∪ B, is the set that contains all elements from both sets A and B.

### Step 3: Determine the maximum cardinality of A ∪ B
The maximum cardinality occurs when there are no common elements between sets A and B. In this case, the cardinality of A ∪ B can be calculated as:
\[ |A ∪ B|_{max} = |A| + |B| = 4 + 3 = 7 \]

### Step 4: Determine the minimum cardinality of A ∪ B
The minimum cardinality occurs when all elements of set B are also elements of set A. In this case, the cardinality of A ∪ B can be calculated as:
\[ |A ∪ B|_{min} = \max(|A|, |B|) = \max(4, 3) = 4 \]

### Step 5: Conclusion
Since we do not have information about the elements of sets A and B, we cannot determine the exact cardinality of A ∪ B. However, we know that:
\[ 4 \leq |A ∪ B| \leq 7 \]
Thus, the exact cardinality of A ∪ B cannot be determined without additional information about the elements in sets A and B.

### Final Answer
The answer to the question is that the exact cardinality of the set A ∪ B cannot be determined.

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