Question 11: Let A = {10, 11, 12, 13} and B = {1, 2, 3, 4}. Write the Following Relations from A to B
Complete Step-by-Step Solution • Exercise 1.3 • Relations • HS First Year Mathematics
Key Takeaways
- Learn how to construct relations between two finite sets.
- Understand relations defined by mathematical conditions.
- Write relations in roster form.
- Identify empty relations correctly.
- Useful for AHSEC, HS First Year, CBSE and other State Board examinations.
Question
Let A = {10, 11, 12, 13} and B = {1, 2, 3, 4} be two sets.
Write the following relations from A to B as defined in each case.
(i) R1 = {(a, b) : a − b is odd, where a ∈ A and b ∈ B}
(ii) R2 = {(a, b) : a + b is a multiple of 4, where a ∈ A and b ∈ B}
(iii) R3 = {(a, b) : a < 10 and a − b ∈ N, where a ∈ A and b ∈ B}
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| Math exercise: relations between sets |
Introduction
A relation from one set to another is a collection of ordered pairs that satisfy a specified condition. To construct a relation, every possible ordered pair is checked against the given rule, and only the valid pairs are included in the final relation.
In this exercise, two finite sets are given, and three different conditions are used to define three relations. We determine each relation step by step in roster form.
Given Sets
Set A
{10, 11, 12, 13}
Set B
{1, 2, 3, 4}
Relations to be Constructed
Relation R1
R1 = {(a, b) : a − b is odd, where a ∈ A and b ∈ B}
Relation R2
R2 = {(a, b) : a + b is a multiple of 4, where a ∈ A and b ∈ B}
Relation R3
R3 = {(a, b) : a < 10 and a − b ∈ N, where a ∈ A and b ∈ B}
Important Concepts
Relation
A subset of the Cartesian product A × B.
Ordered Pair
Each element of a relation is written as (a, b).
Roster Form
A relation written by listing all valid ordered pairs.
Empty Relation
If no ordered pair satisfies the given condition, the relation is the empty set ∅.
Quick Note
To construct a relation, examine every possible ordered pair (a, b) with a ∈ A and b ∈ B. Include only those pairs that satisfy the specified condition.
Solution of Part (i): Relation R1
The given relation is
R1 = {(a, b) : a − b is odd, where a ∈ A and b ∈ B}
Given,
Set A
{10, 11, 12, 13}
Set B
{1, 2, 3, 4}
A difference is odd only when one number is even and the other is odd. Therefore, we check every element of A with every element of B.
Checking the Condition
| Element of A | Valid Elements of B | Reason |
|---|---|---|
| 10 (Even) | 1, 3 | 10 − 1 = 9, 10 − 3 = 7 (Odd) |
| 11 (Odd) | 2, 4 | 11 − 2 = 9, 11 − 4 = 7 (Odd) |
| 12 (Even) | 1, 3 | 12 − 1 = 11, 12 − 3 = 9 (Odd) |
| 13 (Odd) | 2, 4 | 13 − 2 = 11, 13 − 4 = 9 (Odd) |
Construct the Relation R1
Collect all ordered pairs that satisfy the given condition.
R1 = { (10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4) }
Final Answer for R1
| Relation | Answer |
|---|---|
| R1 | { (10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4) } |
Explanation
An even number minus an odd number, or an odd number minus an even number, always gives an odd result. Hence, only these eight ordered pairs satisfy the given condition.
Solution of Part (ii): Relation R2
The given relation is
R2 = {(a, b) : a + b is a multiple of 4, where a ∈ A and b ∈ B}
Given,
Set A
{10, 11, 12, 13}
Set B
{1, 2, 3, 4}
Now check every ordered pair and select only those for which the sum a + b is divisible by 4.
Checking the Condition
| Element of A | Element of B | a + b | Multiple of 4? |
|---|---|---|---|
| 10 | 2 | 12 | ✔ Yes |
| 11 | 1 | 12 | ✔ Yes |
| 12 | 4 | 16 | ✔ Yes |
| 13 | 3 | 16 | ✔ Yes |
All other ordered pairs do not satisfy the given condition.
Construct the Relation R2
Therefore, the required relation is
R2 = {(10,2), (11,1), (12,4), (13,3)}
Final Answer for R2
| Relation | Answer |
|---|---|
| R2 | {(10,2), (11,1), (12,4), (13,3)} |
Explanation
Each ordered pair in R2 satisfies the condition that the sum of its two elements is exactly divisible by 4. Hence, only these four ordered pairs belong to the relation.
Solution of Part (iii): Relation R3
The given relation is
R3 = {(a, b) : a < 10 and a − b ∈ N, where a ∈ A and b ∈ B}
The given sets are
Set A
{10, 11, 12, 13}
Set B
{1, 2, 3, 4}
Check the Given Condition
The first condition is
a < 10
However, every element of set A is
10, 11, 12, 13
None of these elements is less than 10. Therefore, no ordered pair can satisfy the given condition.
Construct the Relation R3
Since there is no valid value of a, the relation contains no ordered pairs.
R3 = ∅
Final Answer for R3
| Relation | Answer |
|---|---|
| R3 | ∅ (Empty Set) |
Explanation
The condition a < 10 is never satisfied because every element of set A is greater than or equal to 10. Hence, the relation contains no ordered pairs and is called the empty relation.
Summary of All Three Relations
| Relation | Result |
|---|---|
| R1 | {(10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4)} |
| R2 | {(10,2), (11,1), (12,4), (13,3)} |
| R3 | ∅ (Empty Set) |
Exam Tips
- ✔ Check every condition carefully before writing ordered pairs.
- ✔ If even one condition is not satisfied, the ordered pair is not included.
- ✔ When no ordered pair satisfies the condition, write the relation as ∅.
- ✔ Always write relations in roster form using curly braces.
- ✔ Do not repeat ordered pairs in a relation.
Frequently Asked Questions (FAQs)
1. What is a relation from A to B?
A relation from set A to set B is any subset of the Cartesian product A × B. It consists of ordered pairs that satisfy a given condition.
2. How is the relation R1 obtained?
In R1, only those ordered pairs are selected for which the difference a − b is an odd number.
R1 = {(10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4)}
3. How is the relation R2 obtained?
In R2, only those ordered pairs are included for which the sum a + b is divisible by 4.
R2 = {(10,2), (11,1), (12,4), (13,3)}
4. Why is R3 an empty relation?
The condition requires a < 10. Since every element of set A is either 10, 11, 12, or 13, no element satisfies the condition.
R3 = ∅
5. What is an empty relation?
An empty relation is a relation that contains no ordered pairs. It is represented by the symbol ∅.
6. What is roster form?
Roster form represents a relation by listing all valid ordered pairs inside curly braces.
Quick Revision
- ✔ A relation is a subset of the Cartesian product.
- ✔ Ordered pairs must satisfy the given condition.
- ✔ R1 contains pairs where a − b is odd.
- ✔ R2 contains pairs where a + b is a multiple of 4.
- ✔ R3 is an empty relation because no element of A is less than 10.
Final Answers
| Part | Answer |
|---|---|
| (i) R1 | {(10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4)} |
| (ii) R2 | {(10,2), (11,1), (12,4), (13,3)} |
| (iii) R3 | ∅ |
Conclusion
In this exercise, we constructed three different relations between the sets A and B by applying the given mathematical conditions. The first relation was obtained by checking whether the difference of the elements was odd, the second by verifying whether the sum was a multiple of four, and the third resulted in an empty relation because none of the elements of set A satisfied the condition a < 10.
Practising questions of this type improves the understanding of relations, ordered pairs, roster form, and logical reasoning, which are essential concepts in higher secondary mathematics.
Final Result
R1 = {(10,1), (10,3), (11,2), (11,4), (12,1), (12,3), (13,2), (13,4)}
R2 = {(10,2), (11,1), (12,4), (13,3)}
R3 = ∅
References
- AHSEC Higher Secondary Mathematics Textbook
- NCERT Mathematics – Relations and Functions
- CBSE Mathematics Curriculum
- Standard Set Theory and Relations Textbooks
- Discrete Mathematics Reference Books