If a set A contains 4 elements, then what is the number of elements in `A xx P(A)`?

Options

A. A) 16
B. B) 32
C. C) 64
D. D) 128

Answer

To solve the problem of finding the number of elements in \( A \times P(A) \) where set \( A \) contains 4 elements, we can follow these steps:

### Step 1: Understand the Power Set
The power set \( P(A) \) of a set \( A \) is the set of all possible subsets of \( A \). If \( A \) has \( n \) elements, the number of elements in the power set \( P(A) \) is given by the formula:
\[ |P(A)| = 2^n \]

### Step 2: Determine the Number of Elements in Set A
In this case, the set \( A \) contains 4 elements. Therefore, we have:
\[ n = 4 \]

### Step 3: Calculate the Number of Elements in the Power Set
Using the formula for the power set:
\[ |P(A)| = 2^4 = 16 \]
So, the power set \( P(A) \) contains 16 elements.

### Step 4: Calculate the Number of Elements in \( A \times P(A) \)
The Cartesian product \( A \times P(A) \) consists of all ordered pairs where the first element is from set \( A \) and the second element is from the power set \( P(A) \). The number of elements in the Cartesian product is calculated as:
\[ |A \times P(A)| = |A| \times |P(A)| \]
Substituting the values we have:
\[ |A| = 4 \quad \text{and} \quad |P(A)| = 16 \]
Thus, we calculate:
\[ |A \times P(A)| = 4 \times 16 = 64 \]

### Final Answer
The number of elements in \( A \times P(A) \) is \( 64 \).

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