If `A={1,2,. . .,60} and B` is relation on A defined as `B={(x,y):y=pq ,"where p and q are primes" ge 3}` then number of elements in B is :

Options

A. 720
B. 660
C. 540
D. 600

Answer

To solve the problem, we need to determine the number of elements in the relation \( B \) defined on the set \( A = \{1, 2, \ldots, 60\} \). The relation \( B \) is defined as \( B = \{(x, y) : y = pq, \text{ where } p \text{ and } q \text{ are primes} \geq 3\} \).

### Step-by-Step Solution:

1. **Identify the Primes Greater Than or Equal to 3**:
The prime numbers greater than or equal to 3 that we will consider are:
\[ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 \]

2. **Calculate Possible Values of \( y \)**:
Since \( y = pq \) must be less than or equal to 60, we will calculate the products of the primes \( p \) and \( q \) and check which products are valid.

3. **List the Valid Combinations**:
- For \( p = 3 \):
- \( q = 3 \): \( 3 \times 3 = 9 \)
- \( q = 5 \): \( 3 \times 5 = 15 \)
- \( q = 7 \): \( 3 \times 7 = 21 \)
- \( q = 11 \): \( 3 \times 11 = 33 \)
- \( q = 13 \): \( 3 \times 13 = 39 \)
- \( q = 17 \): \( 3 \times 17 = 51 \)
- \( q = 19 \): \( 3 \times 19 = 57 \)

- For \( p = 5 \):
- \( q = 5 \): \( 5 \times 5 = 25 \)
- \( q = 7 \): \( 5 \times 7 = 35 \)
- \( q = 11 \): \( 5 \times 11 = 55 \)

- For \( p = 7 \):
- \( q = 7 \): \( 7 \times 7 = 49 \)

- For \( p = 11 \) and higher primes, the products exceed 60, so we stop here.

4. **List of Valid \( y \) Values**:
The valid values of \( y \) that we found are:
\[ 9, 15, 21, 25, 33, 35, 39, 49, 51, 55, 57 \]
This gives us a total of 11 valid \( y \) values.

5. **Determine the Number of Elements in Relation \( B \)**:
For each valid \( y \), \( x \) can be any element from set \( A \) (which has 60 elements). Therefore, for each of the 11 valid \( y \) values, there are 60 possible \( x \) values.

Thus, the total number of elements in the relation \( B \) is:
\[ \text{Total elements in } B = 60 \times 11 = 660 \]

### Final Answer:
The number of elements in the relation \( B \) is \( 660 \).