🔹 1. Introduction
The chapter Relations and Functions forms the foundation of higher mathematics. It connects algebra with real-world mappings and is essential for topics like calculus, trigonometry, and coordinate geometry.
In simple terms:
- A relation shows how elements of one set are connected to elements of another set.
- A function is a special type of relation with stricter rules.
🔹 2. Sets Revision (Important Basics)
Before understanding relations, recall:
✔️ Set
A set is a well-defined collection of objects.
Example:
- A = {1, 2, 3}
- B = {4, 5}
✔️ Types of Sets
- Finite Set
- Infinite Set
- Empty Set (∅)
- Singleton Set
✔️ Cartesian Product
If A and B are sets, then:
A × B = {(a, b) | a ∈ A, b ∈ B}
Example:
If A = {1, 2}, B = {3, 4}
Then:
A × B = {(1,3), (1,4), (2,3), (2,4)}
🔹 3. Relations
✔️ Definition
A relation R from set A to set B is a subset of A × B.
R ⊆ A × B
✔️ Types of Relations
1. Empty Relation
No element is related.
R = ∅
2. Universal Relation
All elements are related.
R = A × B
3. Identity Relation
Every element is related to itself.
R = {(a, a) | a ∈ A}
4. Inverse Relation
If R = {(a, b)}, then
R⁻¹ = {(b, a)}
🔹 4. Relations in a Set
When relation is defined within the same set A:
R ⊆ A × A
✔️ Important Types
1. Reflexive Relation
A relation R is reflexive if:
(a, a) ∈ R for all a ∈ A
Example:
R = {(1,1), (2,2)}
2. Symmetric Relation
If (a, b) ∈ R ⇒ (b, a) ∈ R
Example:
(1,2) and (2,1)
3. Transitive Relation
If (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
4. Equivalence Relation
A relation that is:
- Reflexive
- Symmetric
- Transitive
✔️ Equivalence Class
For element a:
[a] = {x ∈ A | xRa}
Example:
A = {1,2,3}, relation: “same parity”
Classes:
- {1,3}
- {2}
🔹 5. Functions
✔️ Definition
A function f from A to B is a relation such that:
👉 Every element of A has exactly one image in B
Notation:
f: A → B
✔️ Important Terms
Domain
Set of input values (A)
Codomain
Target set (B)
Range
Actual output values
Range ⊆ Codomain
✔️ Example
f(x) = x²
If domain = {1,2,3}
Range = {1,4,9}
🔹 6. Types of Functions
1. One-One Function (Injective)
Different inputs → different outputs
f(a) = f(b) ⇒ a = b
2. Many-One Function
Different inputs → same output
Example:
f(x) = x²
f(2) = 4
f(-2) = 4
3. Onto Function (Surjective)
Every element of codomain is mapped.
Range = Codomain
4. Into Function
Some elements of codomain are not mapped.
5. Bijective Function
Both:
- One-One
- Onto
👉 Important for inverse functions
🔹 7. Algebra of Functions
Let f and g be functions:
✔️ Addition
(f + g)(x) = f(x) + g(x)
✔️ Subtraction
(f − g)(x) = f(x) − g(x)
✔️ Multiplication
(fg)(x) = f(x)g(x)
✔️ Division
(f/g)(x) = f(x)/g(x), g(x) ≠ 0
🔹 8. Composite Functions
Defined as:
(f ∘ g)(x) = f(g(x))
✔️ Example
f(x) = x²
g(x) = x + 1
Then:
(f ∘ g)(x) = (x+1)²
🔹 9. Identity Function
f(x) = x
Denoted by I
🔹 10. Inverse of a Function
If f is bijective, then inverse exists:
f⁻¹(x)
✔️ Finding Inverse
Steps:
- Write y = f(x)
- Swap x and y
- Solve for y
✔️ Example
f(x) = 2x + 3
y = 2x + 3
x = 2y + 3
y = (x – 3)/2
So:
f⁻¹(x) = (x – 3)/2
🔹 11. Graph of Functions
Graphs help visualize functions.
Common graphs:
- Linear: straight line
- Quadratic: parabola
- Modulus: V-shape
🔹 12. Important Functions
1. Polynomial Function
f(x) = a₀ + a₁x + a₂x² + …
2. Rational Function
f(x) = p(x)/q(x)
3. Modulus Function
f(x) = |x|
4. Signum Function
f(x) =
- 1 if x > 0
- 0 if x = 0
- -1 if x < 0
5. Greatest Integer Function
f(x) = [x]
🔹 13. Domain and Range (Important for JEE)
✔️ Finding Domain
Restrictions:
- Denominator ≠ 0
- Square root ≥ 0
- Log argument > 0
✔️ Example
f(x) = 1/(x-2)
Domain: x ≠ 2
✔️ Range
Depends on function behavior.
🔹 14. Binary Operations
A binary operation on set A is:
f: A × A → A
✔️ Properties
- Closure
- Associativity
- Commutativity
- Identity
- Inverse
🔹 15. Important Exam Questions
🔸 Prove relation is equivalence
Check:
- Reflexive
- Symmetric
- Transitive
🔸 Check function type
- One-One test
- Onto test
🔸 Find inverse
Ensure bijection first
🔸 Domain problems (very common in JEE)
🔹 16. JEE & CBSE Important Tips
✔️ Always check domain first
✔️ Learn function transformations
✔️ Practice inverse and composite functions
✔️ Graph understanding is crucial
✔️ Focus on equivalence relations
🔹 17. Common Mistakes
❌ Confusing domain with range
❌ Ignoring restrictions
❌ Assuming all functions are invertible
❌ Not checking bijection before inverse
🔹 18. Quick Revision Formula Sheet
- Relation: R ⊆ A × B
- Function: One input → one output
- Composite: (f ∘ g)(x) = f(g(x))
- Inverse exists ⇔ bijective
- Identity: f(x) = x
🔹 19. Practice Questions
- Check if relation is equivalence
- Find domain of √(x-2)
- Check if function is one-one
- Find inverse of f(x) = (x+1)/(x-1)
- Solve composite functions
🔹 20. Conclusion
Relations and Functions form the backbone of mathematics. A strong grip on this chapter will make future topics like calculus, limits, and differentiation much easier.