4. Z + is the set of positive integers and R: Z+ → Z+ is a relation defined as R = {(a, b) | a, b Є Z+ and a-b >2}. Is it a finite relation? Represent R as a set in the Roster form- Advance Math Class 10- Exercise 1.3

4. Z⁺ is the Set of Positive Integers and R : Z⁺ → Z⁺ is Defined as R={(a,b): a−b>2}. Is it a Finite Relation? Represent R in Roster Form – Advance Math Class 10 Exercise 1.3

Complete NCERT/ASSEB Solution with Step-by-Step Explanation, Roster Form and Finite or Infinite Relation.

📘 Class 10 Mathematics 📚 Exercise 1.3 ⏱ Reading Time: 5 Minutes

$Relation on Positive Integers Exercise 1.3$

Relation Defined on Positive Integers

Introduction

A relation is a collection of ordered pairs satisfying a given condition. In this question, the relation is defined on the set of positive integers using the condition a−b>2. We have to determine whether the relation is finite or infinite and then write the relation in roster form.

Question

Let Z⁺ be the set of positive integers. A relation R : Z⁺ → Z⁺ is defined by

R={(a,b): a,b∈Z⁺ and a−b>2}

Find:

Whether the relation is finite or infinite.
Represent the relation in roster form.

Given Data

Set: Z⁺ = {1,2,3,4,5,...}

Relation:

R={(a,b): a−b>2}

Understanding the Relation

The condition a−b>2 means the first number must be greater than the second number by more than 2.

Examples satisfying the condition are:

(4,1) because 4−1=3>2 ✔
(5,1) because 5−1=4>2 ✔
(5,2) because 5−2=3>2 ✔
(6,1), (6,2), (6,3) ✔
...

Since positive integers never end, infinitely many ordered pairs satisfy the given condition. In the next step, we will write the complete roster form, determine whether the relation is finite or infinite, and present the final answer.

Step-by-Step Solution

The relation is given by

R={(a,b): a,b∈Z⁺ and a−b>2}

Here, Z⁺ denotes the set of all positive integers:

Z⁺={1,2,3,4,5,6,7,8,...}

Step 1: Understand the Given Condition

The condition a−b>2 means that the value of a must be greater than the value of b by more than 2.

Let us examine some values.

a	b	a−b	Belongs to R?
3	1	2	✘ No
4	1	3	✔ Yes
5	1	4	✔ Yes
5	2	3	✔ Yes
6	3	3	✔ Yes
7	4	3	✔ Yes

Step 2: Write the Relation in Roster Form

Collecting all the ordered pairs satisfying the condition, we obtain

R={(4,1),(5,1),(5,2),(6,1),(6,2),(6,3),(7,1),(7,2),(7,3),(7,4),(8,1),(8,2),(8,3),(8,4),(8,5),...}

The dots (...) indicate that the ordered pairs continue forever because there is no largest positive integer.

Step 3: Determine Whether the Relation is Finite

Since the set of positive integers Z⁺ contains infinitely many elements, there are infinitely many ordered pairs satisfying a−b>2.

Therefore, the relation R is an infinite relation.

Final Answer

Roster Form

R={(4,1),(5,1),(5,2),(6,1),(6,2),(6,3),(7,1),(7,2),(7,3),(7,4),(8,1),(8,2),(8,3),(8,4),(8,5),...}

Nature of the Relation

∴ R is an Infinite Relation.

Important Concepts

1. Positive Integers (Z⁺)

The symbol Z⁺ denotes the set of all positive integers:

Z⁺={1,2,3,4,5,6,...}

2. Relation

A relation is a subset of the Cartesian product of two sets that satisfies a specified condition.

3. Roster Form

In roster form, the relation is written by explicitly listing the ordered pairs that satisfy the given condition.

4. Finite Relation

A relation containing a limited number of ordered pairs is called a finite relation.

5. Infinite Relation

A relation containing infinitely many ordered pairs is called an infinite relation.

Quick Revision

✔ Z⁺ represents the set of positive integers.
✔ The condition is a−b > 2.
✔ Every ordered pair satisfying the condition belongs to the relation.
✔ Since positive integers are infinite, the relation also contains infinitely many ordered pairs.
✔ Therefore, the relation is an Infinite Relation.
✔ The roster form is written by listing the ordered pairs followed by "..." to indicate continuation.

Exam Tips

Always understand the given condition before writing the relation.
Write only those ordered pairs that satisfy the condition exactly.
If the underlying set is infinite, check whether the condition produces infinitely many ordered pairs.
Use "..." in roster form whenever the relation continues indefinitely.
State clearly whether the relation is finite or infinite in the final answer.

Frequently Asked Questions (FAQs)

1. What does Z⁺ mean in Mathematics?

Z⁺ denotes the set of all positive integers, namely {1,2,3,4,5,...}.

2. What is a relation?

A relation is a collection of ordered pairs satisfying a given mathematical condition.

3. Why is this relation infinite?

There is no largest positive integer, so infinitely many ordered pairs satisfy the condition a−b > 2.

4. What is roster form?

Roster form represents a relation by listing all valid ordered pairs inside braces.

5. Is Z⁺ a finite set?

No. The set of positive integers is an infinite set.

6. Why are dots (...) used in roster form?

The dots indicate that the pattern continues indefinitely because the relation has infinitely many ordered pairs.

7. What is the final answer to this question?

The relation is an Infinite Relation and its roster form begins as: {(4,1),(5,1),(5,2),(6,1),(6,2),(6,3),...}.

8. Which chapter does this question belong to?

This question belongs to Relations (Exercise 1.3) of Class 10 Mathematics.

References

NCERT Mathematics Textbook – Class 10
ASSEB Mathematics Textbook
Exercise 1.3 – Relations
CBSE Class 10 Mathematics Syllabus
Standard Mathematics Reference Books

Tags: Mathematics