Definition of Relations and Functions in Class 11 with Types Formula and Solved Examples
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 Complete Guide to Concepts and Applications
1. What is a relation in Class 11 Maths?
A relation in Class 11 Maths is a subset of the Cartesian product of two non-empty sets. If A and B are two sets, then any subset of A × B is called a relation from A to B.
- If A = {1, 2} and B = {3, 4}, then A × B = {(1,3), (1,4), (2,3), (2,4)}.
- A relation R could be R = {(1,3), (2,4)}, which is a subset of A × B.
- Relations are usually written as ordered pairs.
2. What is a function in Relations and Functions Class 11?
A function is a special type of relation in which every element of the domain is associated with exactly one element of the codomain. In other words, no input has more than one output.
- If f: A → B, then for each a ∈ A, there exists a unique b ∈ B.
- Example: If A = {1,2,3} and f(x) = x², then f = {(1,1), (2,4), (3,9)}.
- Each input has only one output, so it is a function.
3. What is the difference between relation and function?
The main difference is that a function assigns exactly one output to each input, while a relation may assign multiple outputs to the same input.
- Relation: Any subset of A × B.
- Function: A relation where each element of domain has exactly one image.
- All functions are relations, but not all relations are functions.
4. What is the Cartesian product of two sets?
The Cartesian product of two sets A and B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. It is denoted by A × B.
- If A = {1,2} and B = {3,4},
- Then A × B = {(1,3), (1,4), (2,3), (2,4)}.
- If n(A) = m and n(B) = n, then n(A × B) = m × n.
5. What is domain and range of a function?
The domain is the set of all input values, and the range is the set of all actual output values of a function. The codomain is the set in which outputs lie, but range is the values actually obtained.
- For f(x) = x² where x ∈ {−2, −1, 0, 1, 2}
- Domain = {−2, −1, 0, 1, 2}
- Range = {0, 1, 4}
6. What are the types of functions in Class 11 Maths?
The main types of functions in Class 11 are one-one (injective), onto (surjective), bijective, many-one, and into functions.
- One-one: Different inputs have different outputs.
- Onto: Every element of codomain has a pre-image.
- Bijective: Both one-one and onto.
- Many-one: Different inputs may have same output.
- Into: Some elements of codomain have no pre-image.
7. How do you check if a relation is a function?
A relation is a function if every element of the domain has exactly one image in the codomain. To check:
- Step 1: Look at each first element (input) in ordered pairs.
- Step 2: Ensure no input is repeated with different outputs.
- Example: {(1,2), (2,3), (3,4)} is a function.
- But {(1,2), (1,3)} is not a function.
8. What is a one-one and onto function?
A one-one function maps distinct inputs to distinct outputs, and an onto function covers every element of the codomain.
- One-one (Injective): f(a) = f(b) ⇒ a = b.
- Onto (Surjective): Range = Codomain.
- If a function is both one-one and onto, it is called bijective.
9. What is an identity function in Class 11?
An identity function is a function that maps every element of a set to itself. It is denoted by IA(x) = x for all x ∈ A.
- If A = {1,2,3}, then IA = {(1,1), (2,2), (3,3)}.
- Every element remains unchanged.
- It is always a bijective function.
10. What is the composition of functions?
The composition of functions means applying one function to the result of another and is denoted by (f ∘ g)(x) = f(g(x)).
- First apply g(x).
- Then apply f to the result.
- Example: If f(x) = 2x and g(x) = x + 3, then (f ∘ g)(x) = 2(x + 3) = 2x + 6.
