Definition Types Domain and Range of Relations and Functions with Solved Examples
The concept of relations and functions for class 12 is essential in mathematics and helps in solving board exam and entrance problems efficiently. This topic bridges set theory and algebra, making it foundational for higher mathematics and real-world problems.
Understanding Relations and Functions for Class 12
A relation for class 12 maths represents the association between elements of two sets. If you have two sets, say A and B, a relation from A to B is a subset of their cartesian product, which means it connects some or all pairs where the first element comes from A and the second from B. A function is a special kind of relation—every input from set A relates to exactly one output in set B. This concept is widely used in domain and range questions, set theory, and classification of relations and functions such as one-one, onto, bijective, reflexive, symmetric, and transitive relations.
Difference Between Relation and Function
Many students are confused about the difference between a relation and a function. The table below summarises the key differences:
| Aspect | Relation | Function |
|---|---|---|
| Definition | Any subset of the cartesian product A × B | A relation in which every input has only one output |
| Uniqueness | Not required | Required (no two ordered pairs have the same first element with different second elements) |
| Example | {(1,2), (1,3), (2,4)} | {(1,2), (2,4), (3,5)} |
Understanding this table helps in identifying and working with both relations and functions in board and JEE problems.
Types of Relations and Functions
Relations can be classified as reflexive, symmetric, transitive, and equivalence based on their properties. Functions are further classified as one-one (injective), onto (surjective), many-one, and bijective functions. For example:
1. Reflexive: Every element maps to itself: (a,a) ∈ R for all a ∈ A.2. Symmetric: If (a,b) ∈ R, then (b,a) ∈ R.
3. Transitive: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R.
4. Equivalence: A relation having all three properties above.
For functions, an example of one-one: \( f(x) = x+1 \), onto: \( f(x) = 2x \) when domain and codomain are both all integers.
Key Formulas and Summary Table
Be sure to remember these important formulas for class 12 board revision and JEE:
| Formula / Property | Summary |
|---|---|
| Number of relations from set A to B | If A has m elements, B has n, total relations = \( 2^{mn} \) |
| Number of functions from set A to B | \( n^m \) (A has m elements, B has n) |
| Equivalence Relation | Reflexive + Symmetric + Transitive |
| Composition of functions | \( (g \circ f)(x) = g(f(x)) \) |
| Invertible Function | Bijective (both one-one and onto) |
These formulas are essential for solving relations and functions class 12 problems quickly during revision.
Worked Example – Solving a Relation and Function Problem
Let's solve a typical board level question step by step:
1. Given: Let A = {1,2,3}, B = {2,4}. How many functions can be defined from A to B?2. Step 1: Count the number of elements in set A (m = 3) and set B (n = 2).
3. Step 2: Use the formula for number of functions: \( n^m \)
4. Step 3: Substitute values: \( 2^3 = 8 \)
Therefore, there are 8 possible functions from A to B.
Vedantu recommends always practicing such stepwise questions for sure-shot exam marks.
Practice Problems
- State whether the following relation R on set {1,2,3} given by R = {(1,1), (2,2), (3,3), (1,2), (2,1)} is symmetric.
- If a function f is defined by f(x) = 2x+3, find f(2).
- How many relations can be defined from set P = {a,b} to Q = {1,2,3}?
- Which of the following are functions? {(1,2), (2,3), (2,4)}
Common Mistakes to Avoid
- Confusing relations with functions—remember every function is a relation, but not vice versa.
- Missing out on the need for “unique outputs” for each input in functions.
- Forgetting to check all three properties when asked about equivalence relations.
Real-World Applications
The concept of relations and functions for class 12 appears in computer databases, traffic mapping, social networks, and scientific modeling. Understanding these helps in various engineering and science fields. Vedantu guides students to spot these links between theoretical maths and everyday life.
Suggested Vedantu Learning Links
- Relations and Its Types
- Real Functions
- Composition of Functions and Inverse of a Function
- Cartesian Product and Ordered Pairs
- Onto Function
- Types of Sets
- Difference Between Relation and Function
- Set Theory Symbols
- Union of Sets
- Types of Functions
- Sets
- Relations and Functions Worksheet
We explored the idea of relations and functions for class 12, how to tell the difference, use formulas, apply concepts in problems, and spot them in real life. For more revision and PDF notes, keep practicing with Vedantu’s expert resources and worksheets.
FAQs on Relations and Functions for Class 12 Maths Complete Guide
1. What is a relation in Class 12 Maths?
A relation in Class 12 Maths is a subset of the Cartesian product of two non-empty sets. If A and B are 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 possible relation R = {(1,3), (2,4)} is a subset of A × B.
- Relations are studied under Relations and Functions to understand mappings between elements of sets.
2. What is a function in Relations and Functions?
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 element of the domain has more than one image.
- If f: A → B, then for every a ∈ A, there exists a unique b ∈ B.
- Example: f(x) = x² is a function because each input has one output.
- All functions are relations, but not all relations are functions.
3. What is the difference between a relation and a function?
The main difference is that a function assigns exactly one output to each input, while a relation may assign one or more outputs to an input.
- In a relation, one element of the domain can relate to multiple elements of the codomain.
- In a function, each element of the domain has only one image.
- Thus, every function is a relation, but every relation is not a function.
4. What is a reflexive relation?
A reflexive relation on a set A is a relation in which every element is related to itself. Mathematically, (a, a) ∈ R for all a ∈ A.
- If A = {1, 2}, then R = {(1,1), (2,2)} is reflexive.
- If even one pair (a, a) is missing, the relation is not reflexive.
- Reflexive relations are important in equivalence relations.
5. What is a symmetric relation?
A symmetric relation on a set A is a relation where if (a, b) ∈ R, then (b, a) ∈ R. This means the order of elements can be reversed.
- If (1,2) ∈ R, then (2,1) must also be in R.
- Example: If R = {(1,2), (2,1)}, it is symmetric.
- If any reverse ordered pair is missing, the relation is not symmetric.
6. What is a transitive relation?
A transitive relation on a set A is a relation where if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R. It shows indirect connection.
- If (1,2) and (2,3) are in R, then (1,3) must also be in R.
- If (a, c) is missing, the relation is not transitive.
- Transitivity is a key property in equivalence relations.
7. What is an equivalence relation in Class 12 Maths?
An equivalence relation is a relation that is reflexive, symmetric, and transitive. All three properties must be satisfied.
- Reflexive: (a, a) ∈ R for all a ∈ A.
- Symmetric: If (a, b) ∈ R, then (b, a) ∈ R.
- Transitive: If (a, b) and (b, c) ∈ R, then (a, c) ∈ R.
- Equivalence relations help form equivalence classes.
8. What is a one-one function and onto function?
A one-one (injective) function maps distinct elements of the domain to distinct elements of the codomain, while an onto (surjective) function covers every element of the codomain.
- One-one: If f(a) = f(b), then a = b.
- Onto: For every y in codomain, there exists x in domain such that f(x) = y.
- If both conditions are satisfied, the function is called bijective.
9. How do you find the inverse of a function?
The inverse of a function is found by interchanging x and y and solving for y, provided the function is bijective.
- Step 1: Let y = f(x).
- Step 2: Replace x with y and y with x.
- Step 3: Solve for y.
- Example: If f(x) = 2x + 3, then inverse is f⁻¹(x) = (x − 3)/2.
10. What is the domain and range of a function?
The domain of a function is the set of all possible input values, and the range is the set of all output values actually obtained.
- For f(x) = 1/x, domain is all real numbers except 0.
- For f(x) = x² (x ∈ ℝ), range is all real numbers ≥ 0.
- Finding domain and range is essential in Relations and Functions problems.
