Chapter 2: Relations & Functions

1. Cartesian Product of Sets

If \( A \) and \( B \) are two non-empty sets, the Cartesian product \( A \times B \) is the set of all ordered pairs \( (a, b) \) such that \( a \in A \) and \( b \in B \).

Formula: If \( n(A) = p \) and \( n(B) = q \), then \( n(A \times B) = pq \).

Cartesian Product Engine

Enter elements separated by commas.

Result Set:

Count: pairs

2. Relations

A Relation R from set A to set B is a subset of the Cartesian product \( A \times B \). It is derived by describing a relationship between the first element and the second element.

Types of Relations (Very Important)

Relation Type	Description	Example
Empty (Null)	No ordered pair belongs to the relation.	R = ∅
Universal	Contains all elements of A × B.	R = A × B
Identity	Every element is related to itself.	(a, a) ∀ a ∈ A
Inverse	If (a, b) ∈ R then (b, a) ∈ R⁻¹	R = {(1,2)} ⇒ R⁻¹ = {(2,1)}

Number of Relations (Formula)

If n(A) = p and n(B) = q, then:

Total Relations from A to B = 2^pq

📌 This is because every subset of A × B forms a relation.

Term	Definition
Domain	The set of all first elements of the ordered pairs in R.
Range	The set of all second elements of the ordered pairs in R.
Codomain	The entire set B. (Note: Range \( \subseteq \) Codomain).

Relation Analyzer

Enter relation as pairs like (1,2), (3,4)

Domain:

Range:

3. Functions

A relation \( f \) from set A to set B is said to be a function if every element of set A has one and only one image in set B.

Types of Functions (Mapping Based)

Function Type	Meaning
One–One (Injective)	Different inputs give different outputs.
Many–One	Different inputs may give same output.
Onto (Surjective)	Every element of codomain is an image.
Into	Some elements of codomain are not images.

4. Real Functions & Graphs

Explore standard functions and their graphs.

Graph Gallery

Vertical Line Test

A graph represents a function if and only if any vertical line intersects the graph at most once.

📌 Used to check whether a graph represents a function.

CBSE Tip: If a vertical line cuts the graph at more than one point → NOT a function.

Constant Function

A function of the form f(x) = c, where c is a constant.

Domain = ℝ
Range = {c}

📌 Graph is a straight line parallel to x-axis.

5. Algebra of Functions

For functions \( f: X \to R \) and \( g: X \to R \):

Addition: \( (f+g)(x) = f(x) + g(x) \)
Subtraction: \( (f-g)(x) = f(x) - g(x) \)
Multiplication: \( (fg)(x) = f(x) \cdot g(x) \)
Quotient: \( \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} \), provided \( g(x) \neq 0 \)

Domain Restrictions (Very Important)

For f(x) = 1/x → x ≠ 0
For f(x) = √x → x ≥ 0
For f(x) = 1/√x → x > 0

📌 CBSE often asks domain directly (1 mark).

Concept Mastery Quiz

1. If \( n(A)=3 \) and \( n(B)=2 \), number of relations from A to B is:

2. The domain of the function \( f(x) = \frac{1}{x} \) is:

3. Which ordered pair belongs to \( R = \{(x, y) : y = 2x\} \)?

4. Range of Modulus function \( f(x) = |x| \) is:

5. A function is a special type of: