📌 1. Introduction

In earlier classes, you studied sets and basic functions. In Class 12, the topic is extended to include types of relations, types of functions, and special binary operations.

📖 2. Ordered Pairs

An ordered pair is written as: $(a, b)$ (a,b)

First element → $a$ a
Second element → $b$ b
Order matters: $(a,b) \neq (b,a)$ (a,b)=(b,a)

📊 3. Cartesian Product of Sets

If A and B are two sets, then: $A \times B = \{(a,b) : a \in A, b \in B\}$ A×B={(a,b):a∈A,b∈B}

$A \times B = \{(a,b) : a \in A, b \in B\}$ A×B={(a,b):a∈A,b∈B}

📌 Number of Elements

If:

$n(A) = m$ n(A)=m
$n(B) = n$ n(B)=n

Then: $n(A \times B) = mn$ n(A×B)=mn

🔗 4. Relation

A relation R from set A to set B is a subset of $A \times B$ A×B.

📌 Domain and Range

Domain → First elements
Range → Second elements

📊 5. Types of Relations

1. Empty Relation

No element is related.

2. Universal Relation

Every element is related.

3. Identity Relation

Every element is related to itself.

4. Inverse Relation

If $(a,b) \in R$ (a,b)∈R, then $(b,a) \in R^{-1}$ (b,a)∈R−1

🔁 6. Types Based on Properties

🔹 Reflexive Relation

If: $(a,a) \in R \quad \forall a \in A$ (a,a)∈R∀a∈A

🔹 Symmetric Relation

If: $(a,b) \in R \Rightarrow (b,a) \in R$ (a,b)∈R⇒(b,a)∈R

🔹 Transitive Relation

If: $(a,b) \in R \text{ and } (b,c) \in R \Rightarrow (a,c) \in R$ (a,b)∈R and (b,c)∈R⇒(a,c)∈R

🔹 Equivalence Relation

A relation that is:

Reflexive
Symmetric
Transitive

📌 7. Equivalence Classes

For an element $a$ a, the equivalence class is: $[a] = \{x : xRa\}$ [a]={x:xRa}

$[a] = \{x : xRa\}$ [a]={x:xRa}

🧠 Key Idea:

Equivalence classes divide a set into disjoint subsets.

📖 8. Function

A function is a relation where every element of set A has exactly one image in set B.

📌 Definition:

$f: A \to B$ f:A→B

$f : A \to B$ f:A→B

📊 Domain, Codomain, Range

Domain → Input values
Codomain → Possible outputs
Range → Actual outputs

📊 9. Types of Functions

🔹 One-One Function (Injective)

Different inputs → Different outputs

🔹 Many-One Function

Different inputs → Same output

🔹 Onto Function (Surjective)

Every element of codomain has a pre-image

🔹 Into Function

Some elements of codomain are not mapped

🔹 Bijective Function

Both:

One-One
Onto

👉 Only bijective functions have inverse.

🔁 10. Composition of Functions

If: $f: A \to B, \quad g: B \to C$ f:A→B,g:B→C

Then: $g \circ f (x) = g(f(x))$ g∘f(x)=g(f(x))

$g \circ f(x) = g(f(x))$ g∘f(x)=g(f(x))

🔄 11. Invertible Function

A function is invertible if: $f^{-1}(f(x)) = x$ f−1(f(x))=x

$f^{-1}(f(x)) = x$ f−1(f(x))=x

📌 Condition:

Function must be bijective

🔢 12. Binary Operation

A binary operation on set A is a function: $*: A \times A \to A$ ∗:A×A→A

$* : A \times A \to A$ ∗:A×A→A

📊 Properties of Binary Operations

1. Closure

Result belongs to same set

2. Commutative

$a * b = b * a$ a∗b=b∗a

3. Associative

$(a * b) * c = a * (b * c)$ (a∗b)∗c=a∗(b∗c)

4. Identity Element

$a * e = a$ a∗e=a

5. Inverse Element

$a * a^{-1} = e$ a∗a−1=e

📌 13. Important Results

Every function is a relation
Every relation is not a function
Bijective ⇒ invertible
Composition not commutative

❗ Common Mistakes

Confusing domain and range
Assuming every relation is a function
Forgetting bijective condition for inverse
Errors in composition

🧠 Exam Tips

Practice mapping diagrams
Learn function types clearly
Solve NCERT questions
Focus on properties

📚 Practice Questions

Find domain and range
Check if relation is equivalence
Determine function type
Find inverse function

🎯 Conclusion

Relations and Functions form the backbone of higher mathematics. Understanding these concepts helps in calculus, algebra, and real-life applications.

Mastering this chapter ensures strong conceptual clarity and better performance in exams.