A = {x|x is a prime numbers `le` 100}
B = {x| x is an odd numberc `le` 100}
What is the ratio of the number of subsets of set A to set B?

Options

A. A) `2^(25)`
B. B) `2^(-25)`
C. C) 2
D. D) `((50)^(2))/((25)^(2))`

Answer

To solve the problem, we need to find the ratio of the number of subsets of set A to set B, where:

- Set A contains all prime numbers less than or equal to 100.
- Set B contains all odd numbers less than or equal to 100.

### Step 1: Identify the elements of Set A
Set A consists of prime numbers less than or equal to 100. The prime numbers in this range are:
\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 \]

Counting these, we find that there are 25 prime numbers in total.

### Step 2: Calculate the number of subsets of Set A
The number of subsets of a set with \( n \) elements is given by \( 2^n \). Therefore, for Set A:
\[ n(A) = 25 \]
The number of subsets of Set A is:
\[ \text{Number of subsets of A} = 2^{n(A)} = 2^{25} \]

### Step 3: Identify the elements of Set B
Set B consists of odd numbers less than or equal to 100. The odd numbers in this range are:
\[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99 \]

Counting these, we find that there are 50 odd numbers in total.

### Step 4: Calculate the number of subsets of Set B
Using the same formula for the number of subsets:
\[ n(B) = 50 \]
The number of subsets of Set B is:
\[ \text{Number of subsets of B} = 2^{n(B)} = 2^{50} \]

### Step 5: Calculate the ratio of the number of subsets of Set A to Set B
Now we can find the ratio of the number of subsets of Set A to the number of subsets of Set B:
\[ \text{Ratio} = \frac{\text{Number of subsets of A}}{\text{Number of subsets of B}} = \frac{2^{25}}{2^{50}} = 2^{25 - 50} = 2^{-25} \]

### Final Answer
Thus, the ratio of the number of subsets of Set A to Set B is:
\[ \boxed{2^{-25}} \]