Problem 81 - 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

(D) None of these.

Step 1: Identify the Rational Numbers

We need to find the number of distinct rational numbers in the set \( \frac{p}{q} \) within the interval (0,1). This means \( p < q \). The available integers \( p \) and \( q \) are in the set \( \{1, 2, 3, 4, 5\} \). We'll list all pairs \((p, q)\) where \( p < q \).

Step 2: List Valid Pairs

List all ordered pairs \((p, q)\) such that \( p < q \), where \( p, q \in \{1, 2, 3, 4, 5\} \).- When \( q = 2 \): \((1,2)\)- When \( q = 3 \): \((1,3), (2,3)\)- When \( q = 4 \): \((1,4), (2,4), (3,4)\)- When \( q = 5 \): \((1,5), (2,5), (3,5), (4,5)\)

Step 3: Calculate the Number of Distinct Rationals

Calculate distinct values of \( \frac{p}{q} \):- From \( (1,2) \): \( \frac{1}{2} \)- From \( (1,3) \): \( \frac{1}{3} \)- From \( (2,3) \): \( \frac{2}{3} \)- From \( (1,4) \): \( \frac{1}{4} \)- From \( (2,4) \): \( \frac{1}{2} \) (already counted)- From \( (3,4) \): \( \frac{3}{4} \)- From \( (1,5) \): \( \frac{1}{5} \)- From \( (2,5) \): \( \frac{2}{5} \)- From \( (3,5) \): \( \frac{3}{5} \)- From \( (4,5) \): \( \frac{4}{5} \).Thus, the distinct values are: \[ \{\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}\} \].Thus, \( m = 9 \).

Step 4: Calculate the Number of Mappings

Calculate the number of different mappings from the set \( \{1, 2, 3\} \) onto the set \( \{1, 2\} \). An onto mapping (surjective) ensures each element in \( \{1, 2\} \) is mapped by at least one element from \( \{1, 2, 3\} \). We can have two mappings:- Suppose 2 elements go to 1 and 1 to 2: choose 2 elements to map to 1, 1 element automatically maps to 2. This gives \( \binom{3}{2} = 3 \) configurations.- Similarly, 2 elements going to 2 and 1 to 1 results in \( \binom{3}{2} = 3 \) configurations.Hence, \( n = 3 + 3 = 6 \).

Step 5: Compute m-n

Compute \( m-n = 9 - 6 = 3 \).

Step 6: Answer the Question

Since neither of the provided options matches the computed answer of \( m-n \), \( m-n = 3 \), the correct choice is (D) None of these.

Key Concepts

Rational Numbers

Rational numbers are an important part of algebra and are integral in understanding fractions and relationships between numbers. A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q eq 0 \). In this exercise, the task involves determining distinct rational numbers from the fraction \( \frac{p}{q} \) such that \( p < q \) within the set \( \{1, 2, 3, 4, 5\} \).

• The constraint \( p < q \) ensures that the fraction lies within the interval (0, 1).
• By creating pairs like \((1, 2), (2, 3)\), and so on, we ensure that all possible rational numbers are considered without repetition.
• Each rational is checked for distinctness, so we only count unique values.

By understanding these steps, it becomes clearer how rational numbers are formed and identified, providing a deeper grasp of fractions in the algebraic scope.

Mappings

Mappings in mathematics refer to the process of associating each element of a given set with one or more elements of another set. Specifically, onto mappings, also known as surjective mappings, are those where every element in the target set is mapped by some element in the domain.

• In this example, we map elements from the set \( \{1, 2, 3\} \) onto the set \( \{1, 2\} \).
• An onto mapping ensures that both elements 1 and 2 in the target set are covered by at least one element from the domain set.
• This is achieved by choosing different combinations: either 2 elements map to 1 and 1 to 2, or vice versa.
• The number of such mappings is calculated using combinations, specifically \( \binom{3}{2} \), indicating the number of ways to choose 2 elements out of 3.

These mappings help in understanding functions and their properties, which is a key concept in algebra and other branches of mathematics.

Combinatorics

Combinatorics is a fascinating field of mathematics dealing with counting, arrangement, and combination of objects. In this exercise, combinatorics is used to determine both the number of distinct rational numbers and the number of onto mappings between sets.

• When calculating rational numbers, combinatorics is applied by forming pairs \( \(p, q\) \), where different arrangements of \( p < q \) give distinct fractions.
• For mappings, combinations are essential to figure out how to map elements from one set to another ensuring an onto function.
• Concepts like permutations, combinations, and their respective formulas \( \binom{n}{k} \) ("n choose k") are key in assessing these arrangements.

By mastering combinatorics, students can solve diverse problems dealing with arrangements and probability, which are fundamental skills in both academic and real-world scenarios.