**This note is designed by ‘Basics in Maths’ team. These notes to do help the CBSE 12**^{th }class Maths students fall in love with mathematics and overcome the fear.

^{th }class Maths students fall in love with mathematics and overcome the fear.

**These notes cover all the topics covered in the CBSE 12**^{th }class Maths syllabus and include plenty of formulae and concept to help you solve all the types of12^{th}Math problems asked in the CBSE board and entrance examinations.

^{th }class Maths syllabus and include plenty of formulae and concept to help you solve all the types of12

^{th}Math problems asked in the CBSE board and entrance examinations.

## 1. RELATIONS AND FUNCTIONS

**Ordered pair: **Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

**Cartesian product: **Let A and B are two non- empty sets. The Cartesian product of A and B is denoted by A × B and is defined as set of all ordered pairs (a, b) where a ϵ A and b ϵ B.

**Relation: **Let A and B are two non-empty sets the relation R from A to B is subset of A×B.

⇒ R: A→B is a relation if R⊂ A × B

**Types of relations:**

**Empty relation: – **A relation in a set A is said to be empty relation, if no element of A is related to any element of A.

R = ∅ ⊂ A × A

**Universal Relation: – **A relation in a set A is said to be universal relation, if each element of A is related to every element of A

R = A × A

Both empty relation and universal relation are sometimes called trivial relations.

**Reflexive relation: **A relation R in a set A is said to be reflexive,

if ∀a ∈ A ⇒ (a, a) ∈ R.

**Symmetric relation: **A relation R in a set A is said to be symmetric,

if ∀a, b ∈ A ⇒, (a, b) ∈ R ⇒ (b, a) ∈ R.

**Anti-Symmetric relation: **A relation R in a set A is said to be Anti-symmetric,

if ∀a, b ∈ A; (a, b) ∈ R (b, a) ∈ R ⇒ a = b.

**Transitive relation: **A relation R in a set A is said to be Transitive,

if ∀a, b, c∈ A; (a, b) ∈ R (b, c) ∈ R ⇒ a = c.

**Equivalence relation: **A relation R in a set A is said to equivalence, if it is reflexive, symmetric and transitive.

**Function: **A relation f: X → Y is said to be a function if ∀ xϵ X there exists a unique element y in Y such that (x, y) ϵ f.

(Or)

A relation f: A → B is said to be a function if (i) x ϵ X ⇒ f(x) ϵ Y

(ii) x_{1}, x_{2} ϵ X, x_{1 }= x_{2 }in X ⇒ f(x_{1}) = f(x_{2}) in Y.

** ****TYPES OF FUNCTIONS**

**One– one Function (Injective): – **A function f: X→ Y is said to be one-one function or injective if different elements in X have different images in Y.

(Or)

A function f: X→ Y is said to be one-one function if f(x_{1}) = f(x_{2}) in Y ⇒x_{1} = x_{2} in X.

**On to Function (Surjection): – **A function f: X→ Y is said to be onto function or surjection if for each yϵ Y there exists x ϵ X such that f(x) = y.

** Bijection: – **A function f: X→ Y is said to be Bijection if it is both ‘one-one’ and ‘onto’.

**Composite function: ** If f: A→B, g: B→C are two functions then the composite relation gof is a function from A to C**. **

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).