Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(m=\) number of distinct rational numbers \(\frac{p}{q} \in\) \((0,1)\) such that \(p, q \in\\{1,2,3,4,5\\}\) and \(n=\) number of mappings from \(\\{1,2,3\\}\) onto \(\\{1,2\\}\), then \(m-n\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The answer is (D) None of these.

Step 1: Identify possible fractions

First, determine all possible fractions \( \frac{p}{q} \) where both \( p \) and \( q \) can take values from the set \( \{1, 2, 3, 4, 5\} \). The fractions need to be in the interval \((0, 1)\), so \( p < q \). Count only distinct fractions.

Step 2: Count distinct fractions

The possible \( p < q \) pairs are: \( (1,2), (1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5), (4,5) \). So, the distinct fractions \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{2}{3}, \frac{2}{4} \equiv \frac{1}{2}, \frac{2}{5}, \frac{3}{4}, \frac{3}{5}, \frac{4}{5} \) are created, resulting in \( m = 10 \) distinct rational numbers.

Step 3: Calculate number of mappings

Determine the number of onto mappings from \( \{1, 2, 3\} \) to \( \{1, 2\} \). Use the onto mapping formula for sets: \( 2^n - \binom{n}{1}(2^{n-1}) + \binom{n}{2}(2^{n-2}) \). For this case, there are \( 2^3 - \binom{3}{1}2^2 + \binom{3}{2}2^1 = 8 - 12 + 6 = 2 \) such mappings. Hence, \( n = 6 \).

Step 4: Calculate m-n

With \( m = 10 \) and \( n = 6 \), calculate \( m - n = 10 - 6 = 4 \). Thus, \( m - n \) calculated is 4.

Step 5: Conclusion

Having calculated all necessary steps, find that \( m - n = 4 \), which does not correspond to any of the listed options. Therefore, the answer is (D) None of these.

Key Concepts

Rational Numbers

In mathematics, a rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, where the numerator \( p \) and the denominator \( q \) are integers, and the denominator \( q \) is not zero. They are a fundamental part of arithmetic and algebra.

• For a fraction to represent a rational number, it needs to meet the criterion \( q eq 0 \).
• Rational numbers include integers, finite decimals, and repeating decimals since they can be expressed as fractions.
• In the given exercise, \( p \) and \( q \) are chosen from a finite set \( \{1, 2, 3, 4, 5\} \), with the condition \( p < q \) to ensure the fraction is less than 1.

The exercise focuses on identifying distinct rational numbers within the interval \((0, 1)\). Specifically, it finds unique pairs \((p, q)\) where \( p < q \). This constraint helps in deriving rational numbers that lie strictly between 0 and 1. For example, the fraction \( \frac{2}{4} \) simplifies to \( \frac{1}{2} \), emphasizing the importance of distinct minimal terms.

Mappings

Mappings, particularly from sets, are crucial in understanding functions and relations in mathematics. A mapping from one set to another is a rule that assigns each element from the first set (domain) to an element in the second set (codomain).

• When dealing with mappings from one finite set to another, various functions can be formed, such as one-to-one, onto, and bijective mappings.
• In the context of this exercise, we are interested in onto mappings from the set \( \{1, 2, 3\} \) to \( \{1, 2\} \).

Onto mappings (or surjective functions) are those where every element in the codomain \( \{1, 2\} \) is mapped to by at least one element from the domain \( \{1, 2, 3\} \). Using the formula for onto mappings \[2^n - \binom{n}{1}(2^{n-1}) + \binom{n}{2}(2^{n-2})\]to calculate the number, we find there are two such mappings for the given sets. This is a fascinating application of combinatorial ideas in understanding functions.

Counting Techniques

Counting techniques are essential in combinatorics and help determine the number of ways an event can occur. They are useful in problems involving arrangements, selections, and distributions.

• Combinatorial techniques like permutations and combinations offer solutions for counting different arrangements.
• For the given exercise, these techniques help find the number of distinct rational numbers and mappings.

In the solution, counting techniques help identify 10 distinct rational numbers from the defined conditions by calculating the valid \( (p, q) \) pairs. Similarly, combinatorial logic is applied to compute the number of onto mappings, emphasizing the role of subtraction in combinatorial counting to remove arrangements that do not meet desired criteria.Concepts of combination, often represented by \( \binom{n}{r} \), are vital here for computing such mappings, making the problem-solving strategy systematic and efficient.