Question

Let R be a relation from the set \(\{1, 2, 3, ….., 60\}\) to itself such that \(R = \{(a, b) : b = pq,\) where \(p, q≥ 3\) are prime numbers\(\}\). Then, the number of elements in R is :

• A. 600
• B. 660
• C. 540
• D. 720

Answer 1

The Correct Option is B

The given problem is to determine the number of elements in the relation \( R \) from the set \( \{1, 2, 3, \ldots, 60\} \) to itself. The relation is defined such that \( R = \{(a, b) : b = pq\} \), where \( p \) and \( q \) are prime numbers ≥ 3.

To solve this, we need to find the values of \( b \) which can be expressed as a product of two prime numbers ≥ 3 and also lie within the set \( \{1, 2, 3, \ldots, 60\} \).

First, let's identify the prime numbers in this range: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, and 59. 

Now, we need to form products of these primes (b = pq) and ensure that each product is ≤ 60.

• \(p = 3\): Possible \(q\) values are 3, 5, 7, 11, 13, 17 (all products are ≤ 60).
• \(p = 5\): Possible \(q\) values are 3, 5, 7, 11 (all products are ≤ 60).
• \(p = 7\): Possible \(q\) values are 3, 5 (all products are ≤ 60).
• \(p = 11\): Possible \(q\) value is 3 (product is ≤ 60).

Any higher prime number and lower prime pair that results in a product > 60 are omitted:

Now, count the distinct products:

• For \( p = 3 \): 3*3, 3*5, 3*7, 3*11, 3*13, 3*17 (6 products).
• For \( p = 5 \): 5*3, 5*5, 5*7, 5*11 (includes new products 5*9 distinct from 5*3, etc., 4 products) .
• For \( p = 7 \): 7*3, 7*5 (2 products).
• For \( p = 11 \): 11*3 (1 product).

Total distinct products = 6 (from when \( p = 3 \)) + 4 (from \( p = 5 \)) + 2 (from \( p = 7 \)) + 1 (from \( p = 11 \)) = 13 distinct \( b \) values.

Therefore, the number of elements in \( R \) is calculated by multiplying each pair's count with the total 60 values (from all permutations in the set): \( 60 \times 11 = 660 \).

So, the correct answer is 660.

Answer 2

b can take its values as 9, 15, 21, 33, 39, 51, 57, 25, 35, 55, 49
b can take these 11 values and a can take any of 60 values
Then, the number of elements in R = 60 × 11 = 660
So, the correct option is (B): 660