Problem 9 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A\) is the set of even natural numbers less than 8 and \(\mathrm{B}\) is the set of prime numbers less then 7 , then the number of relations from \(A\) to \(B\) is (A) \(2^{9}\) (b) \(9^{2}\) (C) \(3^{2}\) (D) \(2^{9}-1\)

Step-by-Step Solution

Answer

The number of relations from set A to set B is \(2^9\), which is option (A).

Step 1: Identify Set A

Set \( A \) is the set of even natural numbers less than 8. These numbers are 2, 4, and 6. This means \( A = \{ 2, 4, 6 \} \).

Step 2: Identify Set B

Set \( B \) consists of prime numbers less than 7. These numbers are 2, 3, and 5. Therefore, \( B = \{ 2, 3, 5 \} \).

Step 3: Determine the Number of Possible Relations

The number of relations from one set to another set is given by \( 2^{|A| \times |B|} \), where \(|A|\) and \(|B|\) are the numbers of elements in sets \(A\) and \(B\) respectively.

Step 4: Calculate \(|A|\) and \(|B|\)

The number of elements in set \(A\) is three (2, 4, 6), so \(|A| = 3\). The number of elements in set \(B\) is also three (2, 3, 5), hence \(|B| = 3\).

Step 5: Final Calculation: Number of Relations

Using the formula, the number of possible relations is \( 2^{|A| \times |B|} = 2^{3 \times 3} = 2^9 \).

Step 6: Select the Correct Option

The number of relations calculated is \( 2^9 \), which corresponds to option (A). Therefore, the correct answer is (A) \(2^9\).

Key Concepts

Even Natural Numbers

In mathematics, natural numbers are positive integers starting from 1. When we talk about *even* natural numbers, we refer to those natural numbers that are divisible by 2.
These numbers are in the form of 2, 4, 6, 8, etc. They can be generated by the formula:

• Even number = 2n, where n is a whole number.

In the context of our exercise, we need to focus only on even natural numbers less than 8. So, our set of even natural numbers would be {2, 4, 6}.
These are the numbers that fit both criteria of being even and being less than 8. Understanding this will help in identifying elements of the set A clearly.

Prime Numbers

Prime numbers are numbers greater than 1, which are only divisible by themselves and 1. They cannot be divided evenly by any other integers.
This makes them quite unique and serves as building blocks in number theory. For instance, the number 2 is special as it is the only even prime number.
The number 3 can only be divided by 1 and 3, and the number 5 by 1 and 5. Thus, they are all prime. For this task, we focus on prime numbers less than 7. So, our set of relevant prime numbers would be {2, 3, 5}.

Sets and Subsets

The concept of sets is fundamental in mathematics. Sets are collections of distinct objects, considered as an object themselves. These objects are called elements or members of the set.
When dealing with sets, it's important to understand that sets can be grouped to form subsets. A subset is a set whose elements are all contained within another set. For example, if we have a set A = {2, 4, 6}, then:

• {2, 4} is a subset of A.
• {6} is also a subset of A.
• The set {2, 3} is not a subset of A, since 3 is not contained within A.

Working with sets requires clear definitions of the collection's elements, which in this case, helps us in establishing relationships between sets A and B.

Calculation of Relations

In set theory, a relation from set A to set B is a subset of the Cartesian product of A and B. The Cartesian product, denoted as A × B, is the set of all possible ordered pairs where the first element is from A and the second element is from B.
If |A| = m and |B| = n, then the Cartesian product A × B has mn possible pairs. The number of possible relations from A to B is given by the formula:

• 2^(m×n)

In our exercise, both sets A and B have 3 elements each, which makes |A × B| = 3 × 3 = 9 . Therefore, there are 2^9 possible relations.
Applying this formula ensures accuracy when figuring out the number of relations between finite sets.