If A = (1, 3, 5, 7), then what is the cardinality of the power set P(A)?

Options

A. A)8
B. B)15
C. C)16
D. D)17

Answer

To find the cardinality of the power set \( P(A) \) for the set \( A = \{1, 3, 5, 7\} \), we can follow these steps:

### Step 1: Determine the cardinality of set \( A \)
The cardinality of a set is the number of elements it contains. In this case, the set \( A \) has the elements \( 1, 3, 5, \) and \( 7 \).

- **Elements in set \( A \)**: 1, 3, 5, 7
- **Number of elements**: 4

Thus, the cardinality of set \( A \) is \( n(A) = 4 \).

### Step 2: Use the formula for the cardinality of the power set
The cardinality of the power set \( P(A) \) can be calculated using the formula:
\[ N(P(A)) = 2^n \]
where \( n \) is the cardinality of the original set \( A \).

### Step 3: Substitute the value of \( n \)
We have already determined that \( n(A) = 4 \). Now we can substitute this value into the formula:
\[ N(P(A)) = 2^4 \]

### Step 4: Calculate \( 2^4 \)
Now we calculate \( 2^4 \):
\[ 2^4 = 16 \]

### Conclusion
The cardinality of the power set \( P(A) \) is \( 16 \).

### Final Answer
The cardinality of the power set \( P(A) \) is \( 16 \).

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