Question 12: Find the Domain and Range of the Relations R1 and R2 | Exercise 1.3
Complete Step-by-Step Solution • Relations • HS First Year Mathematics
Key Takeaways
- Learn how to determine the domain and range of a relation.
- Understand relations defined using set-builder notation.
- Convert relations into roster form.
- Find the domain and range of finite and infinite relations.
- Useful for AHSEC, HS First Year, CBSE and other State Board examinations.
Question
R1 and R2 are two relations on the set of integers Z defined as follows:
(i) R1 = {(a, b) : a2 = b2, a, b ∈ Z}
(ii) R2 = {(a, b) : a − b is positive, where a and b are prime numbers smaller than 10}
Determine the Domain and the Range of each relation.
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| Math relations on integers |
Introduction
A relation is a collection of ordered pairs that connects elements of one set with elements of another set (or the same set). Two important concepts associated with every relation are its domain and range.
In this exercise, two different relations are defined using set-builder notation. We first convert each relation into an understandable form and then determine its domain and range step by step.
Given Relations
Relation R1
R1 = {(a, b) : a2 = b2, a, b ∈ Z}
Relation R2
R2 = {(a, b) : a − b > 0, where a and b are prime numbers less than 10}
Important Definitions
Relation
A relation is a set of ordered pairs satisfying a given condition.
Domain
The domain is the set of all first elements of the ordered pairs.
Range
The range is the set of all second elements of the ordered pairs.
Set Builder Form
A mathematical notation used to describe a set by specifying a rule or condition.
Quick Note
To determine the domain, collect every first element of the ordered pairs. To determine the range, collect every second element. Repeated elements are written only once because a set never contains duplicate elements.
Solution of Part (i): Relation R1
The given relation is
R1 = {(a, b) : a2 = b2, a, b ∈ Z}
Since a2 = b2, the possible values of b are either a or -a. Therefore,
a = b
or
a = -b
Understanding the Relation
Some ordered pairs belonging to the relation are:
(0,0), (1,1), (1,-1), (-1,1), (-1,-1),
(2,2), (2,-2), (-2,2), (-2,-2), ...
Since the relation is defined on the set of all integers, infinitely many ordered pairs satisfy the given condition.
Find the Domain of R1
The domain is the set of all first components of the ordered pairs.
d(R1) = Z
Every integer can appear as the first element of an ordered pair satisfying a2 = b2.
Find the Range of R1
The range is the set of all second components of the ordered pairs.
r(R1) = Z
Every integer can also appear as the second element because for every b ∈ Z, there exists an integer a such that a = b or a = -b.
Result for Relation R1
| Particular | Answer |
|---|---|
| Relation | a2 = b2 |
| Domain | Z |
| Range | Z |
Conclusion
Since every integer satisfies the relation with either itself or its additive inverse, both the domain and the range of R1 are the complete set of integers.
Solution of Part (ii): Relation R2
The given relation is
R2 = {(a, b) : a − b > 0, where a and b are prime numbers less than 10}
First, write all the prime numbers less than 10.
{2, 3, 5, 7}
Construct the Relation R2
The condition a − b > 0 means that a > b. Now check every possible ordered pair.
| a | Possible b | Ordered Pairs |
|---|---|---|
| 2 | None | — |
| 3 | 2 | (3,2) |
| 5 | 2, 3 | (5,2), (5,3) |
| 7 | 2, 3, 5 | (7,2), (7,3), (7,5) |
Hence,
R2 = {(3,2), (5,2), (5,3), (7,2), (7,3), (7,5)}
Find the Domain of R2
The domain is the set of all first elements of the ordered pairs.
d(R2) = {3, 5, 7}
Find the Range of R2
The range is the set of all second elements of the ordered pairs.
r(R2) = {2, 3, 5}
Result for Relation R2
| Particular | Answer |
|---|---|
| Prime Numbers Used | {2, 3, 5, 7} |
| Relation R2 | {(3,2), (5,2), (5,3), (7,2), (7,3), (7,5)} |
| Domain | {3, 5, 7} |
| Range | {2, 3, 5} |
Observation
The smallest prime number 2 cannot appear in the domain because there is no smaller prime number less than 2. Similarly, the largest prime number 7 cannot appear in the range because no larger prime number less than 10 satisfies the given condition.
Summary of Both Relations
After evaluating both relations, we obtain the following domains and ranges.
| Relation | Domain | Range |
|---|---|---|
| R1 = {(a, b) : a2 = b2, a, b ∈ Z} | Z | Z |
| R2 = {(a, b) : a − b > 0} | {3, 5, 7} | {2, 3, 5} |
Why Are These Answers Correct?
Relation R1
Since every integer has the same square as itself and its additive inverse, every integer appears both as a first element and as a second element.
Relation R2
Only ordered pairs satisfying a > b are included. Therefore, the smallest prime never appears in the domain, while the largest prime never appears in the range.
Important Concepts
Domain
The set of all first elements of the ordered pairs.
Range
The set of all second elements of the ordered pairs.
Relation
A subset of the Cartesian product that satisfies a given condition.
Roster Form
A method of writing a relation by explicitly listing all ordered pairs.
Common Mistakes Students Make
❌ Mistake 1
Confusing the domain with the range.
❌ Mistake 2
Repeating the same element more than once while writing a set.
❌ Mistake 3
Ignoring the given condition before writing the ordered pairs.
❌ Mistake 4
Using composite numbers instead of only the given prime numbers.
Exam Tips
- ✔ Write all ordered pairs first before finding the domain and range.
- ✔ Remember that a set never contains duplicate elements.
- ✔ The domain always consists of the first coordinates of the ordered pairs.
- ✔ The range always consists of the second coordinates of the ordered pairs.
- ✔ Read the given condition carefully before constructing the relation.
- ✔ For finite sets, prepare the ordered pairs systematically to avoid missing any valid pair.
Memory Tricks
- 🟢 Domain → D comes first → First elements.
- 🔵 Range → Remaining (second) elements.
- 🟢 Relation = Collection of ordered pairs.
- 🔵 Always remove repeated elements while writing sets.
Frequently Asked Questions (FAQs)
1. What is a relation in mathematics?
A relation is a collection of ordered pairs that connects elements of one set with elements of another set (or the same set) according to a specific rule.
2. What is the domain of a relation?
The domain is the set of all first elements of the ordered pairs in a relation.
3. What is the range of a relation?
The range is the set of all second elements of the ordered pairs in a relation.
4. What is the domain and range of R1?
d(R1) = Z
r(R1) = Z
Since every integer has the same square as itself and its additive inverse, every integer appears in both the domain and the range.
5. What is the domain and range of R2?
d(R2) = {3, 5, 7}
r(R2) = {2, 3, 5}
6. Why is 2 not included in the domain of R2?
The relation requires a − b > 0. Since there is no prime number smaller than 2, it cannot appear as the first element of any ordered pair.
7. Why is 7 not included in the range of R2?
There is no prime number less than 10 that is greater than 7. Hence, 7 cannot appear as the second element of any ordered pair satisfying the given condition.
8. What is roster form?
Roster form represents a relation by listing all valid ordered pairs inside curly braces.
Quick Revision
- ✔ Domain = Set of first elements.
- ✔ Range = Set of second elements.
- ✔ R1 is defined on all integers.
- ✔ d(R1) = Z and r(R1) = Z.
- ✔ Prime numbers less than 10 are {2, 3, 5, 7}.
- ✔ d(R2) = {3, 5, 7}.
- ✔ r(R2) = {2, 3, 5}.
Final Answers
| Relation | Domain | Range |
|---|---|---|
| R1 | Z | Z |
| R2 | {3, 5, 7} | {2, 3, 5} |
Final Result
For R1:
d(R1) = Z
r(R1) = Z
For R2:
d(R2) = {3, 5, 7}
r(R2) = {2, 3, 5}
Conclusion
This exercise demonstrates how to determine the domain and range of relations defined using mathematical conditions. For R1, every integer satisfies the relation with itself or its additive inverse, so both the domain and range are the complete set of integers. For R2, only prime numbers less than 10 that satisfy the condition a > b are included, resulting in finite domain and range sets.
Understanding these concepts is essential for higher mathematics because domain and range form the basis for studying functions, mappings, and advanced relations in later chapters.
References
- AHSEC Higher Secondary Mathematics Textbook
- NCERT Mathematics – Relations and Functions
- CBSE Mathematics Curriculum
- Standard Set Theory and Relations Textbooks
- Discrete Mathematics Reference Books