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 :
- 600
- 660
- 540
- 720
The Correct Option is B
Approach Solution - 1
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.
Approach Solution -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
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Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
