8. A = {1,2,3,4} and B = {a,b,c,d} Be Two Sets. Choose Which of the Following Are Relations from A to B – Advance Math Class 10 Exercise 1.3
Step-by-Step NCERT Solution with Complete Explanation of Relations, Cartesian Product, and Identification of Valid Relations.
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| Relation from Set A to Set B |
Introduction
A relation from one set to another is any subset of the Cartesian product of the two sets. Therefore, every ordered pair in the relation must have its first element from set A and its second element from set B. If even one ordered pair does not belong to the Cartesian product, then the given collection cannot be considered a relation from A to B.
In this exercise, we are given two finite sets and several collections of ordered pairs. Our task is to determine which of them are valid relations from A to B by comparing each ordered pair with the Cartesian product A × B.
Question
Let
A = {1,2,3,4}
B = {a,b,c,d}
Choose which of the following collections are relations from A to B.
Given Data
- Set A = {1,2,3,4}
- Set B = {a,b,c,d}
- We have to examine each collection of ordered pairs.
- An ordered pair belongs to a relation only if it belongs to A × B.
Cartesian Product A × B
The Cartesian product of sets A and B is the set of all possible ordered pairs whose first element belongs to A and second element belongs to B.
A × B =
{(1,a), (1,b), (1,c), (1,d), (2,a), (2,b), (2,c), (2,d), (3,a), (3,b), (3,c), (3,d), (4,a), (4,b), (4,c), (4,d)}
Step-by-Step Solution
A collection of ordered pairs is a relation from A to B only when every ordered pair belongs to the Cartesian product A × B. Now examine each case individually.
(i) {(1,a), (1,b), (2,c), (4,d)}
Verification
- (1,a) ∈ A × B ✔
- (1,b) ∈ A × B ✔
- (2,c) ∈ A × B ✔
- (4,d) ∈ A × B ✔
Answer: YES
Every ordered pair belongs to A × B. Hence, it is a relation from A to B.
(ii) {(1,1), (1,a), (3,c)}
Verification
- (1,1) ✖
- (1,a) ✔
- (3,c) ✔
The second element of the ordered pair (1,1) does not belong to set B. Therefore this ordered pair is not an element of A × B.
Answer: NO
Hence, this collection is not a relation from A to B.
(iii) A × B
The Cartesian product A × B itself contains every possible ordered pair whose first element belongs to A and second element belongs to B.
Answer: YES
Since every ordered pair belongs to A × B, it is itself a valid relation from A to B.
(iv) {(a,1), (b,2), (c,3)}
Verification
Here the first elements are a, b, c, which belong to set B, whereas the first element of every ordered pair should belong to set A.
Answer: NO
Therefore, this collection is not a relation from A to B.
(v) {∅}
The element of this set is ∅, which is not an ordered pair belonging to A × B.
Answer: NO
Hence, {∅} is not a relation from A to B.
(vi) {(1,c), (2,c), (3,c), (4,c)}
Verification
- (1,c) ✔
- (2,c) ✔
- (3,c) ✔
- (4,c) ✔
Answer: YES
All the ordered pairs belong to A × B. Hence, this is a valid relation from A to B.
Final Answer
| Case | Relation? | Reason |
|---|---|---|
| (i) | ✔ Yes | All ordered pairs belong to A × B. |
| (ii) | ✘ No | (1,1) is not an element of A × B. |
| (iii) | ✔ Yes | A × B itself is a relation. |
| (iv) | ✘ No | First elements belong to B instead of A. |
| (v) | ✘ No | {∅} is not a set of ordered pairs. |
| (vi) | ✔ Yes | Every ordered pair belongs to A × B. |
Important Concepts
1. Relation
A relation from set A to set B is any subset of the Cartesian product A × B.
2. Cartesian Product
The Cartesian product contains all possible ordered pairs whose first element belongs to A and second element belongs to B.
3. Ordered Pair
An ordered pair (x,y) belongs to A × B only if x ∈ A and y ∈ B.
4. Valid Relation
Every ordered pair of the given collection must belong to A × B. If even one ordered pair does not belong to A × B, the collection is not a relation from A to B.
Quick Revision
- ✔ Relation = Subset of A × B.
- ✔ First element must belong to set A.
- ✔ Second element must belong to set B.
- ✔ Every ordered pair should satisfy the definition of Cartesian product.
- ✔ A × B itself is always a relation from A to B.
Exam Tips
- Always write the Cartesian product before checking any relation.
- Check every ordered pair individually.
- If one ordered pair is invalid, the entire collection is not a relation.
- Do not confuse {∅} with the empty relation ∅. The empty relation ∅ is a valid relation, whereas {∅} is not a relation because its only element is not an ordered pair.
- Remember that every subset of A × B is a relation.
Frequently Asked Questions (FAQs)
1. What is a relation from A to B?
A relation from A to B is any subset of the Cartesian product A × B.
2. What is A × B?
It is the set of all ordered pairs whose first element belongs to set A and second element belongs to set B.
3. Why is (1,1) not in A × B?
Because the second element 1 does not belong to set B = {a,b,c,d}.
4. Is A × B itself a relation?
Yes. Since every relation is a subset of A × B, the Cartesian product itself is also a valid relation.
5. Why is {(a,1), (b,2), (c,3)} not a relation?
The first elements belong to set B instead of set A, so these ordered pairs are not elements of A × B.
6. Is the empty relation valid?
Yes. The empty relation ∅ is always a valid relation because it is a subset of every Cartesian product. However, {∅} is not a relation since its only element is not an ordered pair.
7. Which options are correct in this question?
The valid relations are (i), (iii), and (vi).
Related Questions – Exercise 1.3
1. Identity Relation and Universal Relation on A
2. Write Down All Relations on A
3. Find the Elements of the Relation R
4. Represent the Relation in Roster Form
5. Relation in Tabular Form, Arrow Diagram and Matrix
6. Determine R-1, Domain and Range
7. Draw Arrow Diagram and Find Domain and Range
9. Relation R Defined on Natural Numbers
10. Relation on A Defined by Divisibility
References
- NCERT Mathematics – Class 10
- ASSEB (SEBA) Mathematics Textbook
- Relations and Functions – Exercise 1.3
- CBSE Mathematics Curriculum
- Standard School Mathematics Reference Books