Difference Between Relation and Function with Examples and Table
The concept of Relations and Functions Class 11 is essential in mathematics and helps students understand how to connect two sets, solve mapping problems, and build a strong foundation for board exams and competitive tests.
Understanding Relations and Functions Class 11
A relation is defined as a subset of the cartesian product of two sets and describes how elements from one set are related to elements from another set. A function is a special type of relation, in which every element of the first set (domain) is related to exactly one element of the second set (co-domain). This concept is widely used in real-valued functions, domain and range determination, and composition and inverse problems.
Basic Concepts and Definitions
1. Set: A collection of distinct objects, e.g., A = {1, 2, 3}.
2. Cartesian Product: For sets A and B, A × B = { (a, b): a ∊ A, b ∊ B }. All possible ordered pairs.
3. Relation: Any subset of the cartesian product A × B.
4. Domain: Set of all first elements of the ordered pairs in relation.
5. Range: Set of all second elements of the ordered pairs in relation.
6. Function: A relation in which each input has exactly one output, i.e., for each a ∊ A, there is only one b ∊ B such that (a, b) ∊ f.
Formulae Used in Relations and Functions Class 11
The standard notation for a function is: \( f: A \rightarrow B \), where for each \( a \in A \), \( \exists! \ b \in B \) such that \( f(a) = b \).
The number of relations from a set A (with m elements) to a set B (with n elements) is \( 2^{mn} \).
Here’s a helpful table to understand relations and functions more clearly:
Difference Between Relation and Function
| Basis | Relation | Function |
|---|---|---|
| Definition | Any subset of A × B | Assigns exactly one output in B to each input in A |
| Uniqueness | Each input can have multiple outputs | Each input has only one output |
| Diagram | May have lines splitting from one input to many outputs | One arrow from input to only one output |
This table shows the fundamental differences which are tested frequently in exams.
Types of Relations and Functions with Examples
| Type | Definition | Example |
|---|---|---|
| Reflexive Relation | (a, a) ∈ R for all a ∈ A | “Is equal to” |
| Symmetric Relation | If (a, b) ∈ R ⇒ (b, a) ∈ R | “Is sibling of” |
| Transitive Relation | (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R | “Is ancestor of” |
| Injective (One-to-one) | Each output matches only one input | Click for details |
| Surjective (Onto) | Every element in co-domain is an output | Click for examples |
| Bijective | Both injective and surjective | f(x) = x (identity mapping) |
Worked Example – Solving a NCERT Type Problem
Example: Let X = {a, b, c} and Y = {1, 2, 3}. Find the number of relations from X to Y.
1. Find cartesian product: X × Y = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3), (c,1), (c,2), (c,3)}
2. Count elements in X × Y: There are 3 × 3 = 9 ordered pairs.
3. Number of possible relations = Number of subsets of X × Y = 2⁹ = 512.
Final Answer: 512 relations.
Practice Questions
1. Give an example of a relation that is not a function from A = {1,2} to B = {3,4}.
2. For A = {a, b} and B = {x, y}, list all possible functions from A to B.
3. State whether the following relation is a function: R = {(1,2), (2,2), (1,3)}.
4. Fill in: The number of relations from a set with 3 elements to a set with 2 elements = ?
Common Mistakes to Avoid
- Assuming every relation is a function.
- Mixing up domain, co-domain, and range.
- Forgetting that in a function, each input can only have one output.
- Confusing onto (surjective) and one-to-one (injective) functions—remember to check both conditions.
Real-World Applications
The concept of relations and functions is crucial in coding (mapping user input to outputs), database search logic, social media connections (friend recommendations), and more. Vedantu lessons help students visualise these relationships through diagrams and real-life mapping problem sets.
Quick Revision Notes and Worksheets
For last-minute revision, it is handy to remember: 1. A function cannot assign two different outputs to the same input. 2. Shortcuts: In sets of m and n elements, number of relations = \( 2^{mn} \); number of functions = \( n^m \). Download worksheets for extra practice: Relations and Functions Worksheet.
Suggested Internal Links for Further Study
- Basics of Relations and Its Types
- Domain and Range Relations
- Real Functions
- Composition of Functions and Inverse of a Function
- One-to-One Function
- Onto Function
- Types of Sets
- Functions and Its Types
- Difference Between Relation and Function
- Sets, Relations, and Functions Important Questions
- Union of Sets
We explored the idea of relations and functions Class 11, how to distinguish between them, solve problems using stepwise methods, and their importance in both mathematics and daily life. Practice more with Vedantu to master these concepts and excel in your exams.
FAQs on Relations and Functions - Class 11 Maths Explained
1. What is a relation in Class 11 Maths?
In Class 11 Maths, a relation is defined as a subset of the Cartesian product of two non-empty sets. It represents a connection between elements of these sets, where each ordered pair signifies how the first element is related to the second.
2. What is a function? How to identify a function?
A function is a special type of relation where every element in the domain (first set) maps to exactly one element in the range (second set). To identify a function, check that for every input there is only one output. Function notation is written as f(x) = y, indicating that y is the image of x under the function f.
3. Difference between relation and function with example?
The key difference is that a relation may associate an input with multiple outputs, while a function assigns each input exactly one output. Example: Relation R = {(1, 2), (1, 3)} is not a function because input 1 has two outputs (2 and 3). However, function f = {(1, 2), (2, 3)} is a function because each input corresponds to only one output.
4. What are the types of functions in Class 11?
Class 11 learners study several important types of functions including:
• One-to-One (Injective): Each element of the domain maps to a unique element of the range.
• Onto (Surjective): Every element of the range has at least one pre-image.
• Bijective: Both one-to-one and onto, establishing a perfect pairing.
• Constant Function, Identity Function, Polynomial Function, etc. Each has distinct properties important for exams.
5. Where to find relations and functions class 11 NCERT solutions?
The relations and functions Class 11 NCERT solutions are available on several educational portals including Vedantu, BYJU’S, and the official NCERT website. These solutions provide stepwise answers to textbook problems and miscellaneous exercises, helping students prepare effectively for board exams.
6. List some real-life examples of relations and functions?
Real-life examples of relations and functions include:
• Assigning students to their roll numbers (function).
• Mapping employees to their departments (relation).
• Calculating the square of a number (function).
• Pairing customers with their ordered products (relation).
These scenarios help in understanding abstract concepts practically.
7. Why do many students confuse mapping diagrams for functions?
Students often confuse mapping diagrams because they may not carefully check for multiple arrows from one input to different outputs. Remember, for a diagram to represent a function, each input should point to exactly one output. Overlapping or multiple arrows violate this, indicating only a relation, not a function.
8. How can a relation not be a function? Give a counterexample.
A relation fails to be a function if any element in the domain corresponds to more than one element in the range. Counterexample: Relation R = {(3,4), (3,5)} is not a function because the input 3 relates to outputs 4 and 5, violating the function rule of unique outputs.
9. Why is function notation (f(x)) important in exams?
Function notation f(x) is important because it clearly expresses the output value for every input in a standardized, concise form. It simplifies working with functions in algebraic manipulations, composition, and inverse functions, which are frequently tested in Class 11 and competitive exams.
10. What’s the fastest way to check if a relation is a function?
The quickest method is to verify that each input corresponds to exactly one output by examining ordered pairs or mapping diagrams. If any input repeats with different outputs, the relation is not a function. This rule is often used in multiple-choice questions for quick elimination.
11. Why do functions matter in calculus later?
Functions are foundational to calculus because calculus studies how functions change and behave. Concepts like limits, derivatives, and integrals analyze functions’ rates of change and areas under curves, making mastery of functions essential for understanding higher mathematics.
