Relations and Functions for Class 12 Maths

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.