Chapter 1 Relations and Functions

Notes

Relations and Functions One Shot Notes

Cartesian Product

Agar A aur B do sets hain, to A × B ka matlab hai ordered pairs ka set.

A × B = {(a,b) : a ∈ A and b ∈ B}

Example:

A = {1, 2}

B = {3, 4}

A × B = {(1,3), (1,4), (2,3), (2,4)}

Formula:

n(A × B) = n(A) × n(B)

Important:

A × B generally B × A ke equal nahi hota.

Ordered Pair

Ordered pair ka form hota hai:

(a,b)

Yaha order important hota hai.

Example:

(1,2) ≠ (2,1)

Relation

A se B tak relation, A × B ka subset hota hai.

R ⊆ A × B

Example:

A = {1, 2}

B = {3, 4}

R = {(1,3), (2,4)}

Ye relation hai.

Domain

Relation ke first elements ka set domain hota hai.

Example:

R = {(1,2), (3,4), (5,6)}

Domain = {1, 3, 5}

Range

Relation ke second elements ka set range hota hai.

Example:

R = {(1,2), (3,4), (5,6)}

Range = {2, 4, 6}

Codomain

Jis set me output expected hota hai, use codomain bolte hain.

Important:

Range codomain ka subset hota hai.

Range ⊆ Codomain

Types of Relations

Empty Relation:

Jisme koi ordered pair nahi hota.

Example:

R = {}

Universal Relation:

Jisme saare possible ordered pairs hote hain.

Identity Relation:

Jisme har element apne aap se related hota hai.

Example:

A = {1, 2, 3}

I = {(1,1), (2,2), (3,3)}

Reflexive Relation

Relation R set A par reflexive tab hota hai jab har a ∈ A ke liye (a,a) ∈ R ho.

Simple trick:

Har element ka self-pair hona chahiye.

Example:

A = {1, 2}

R = {(1,1), (2,2), (1,2)}

Ye reflexive hai.

Symmetric Relation

Relation symmetric tab hota hai jab:

Agar (a,b) ∈ R, to (b,a) bhi R me hona chahiye.

Example:

R = {(1,2), (2,1)}

Ye symmetric hai.

Transitive Relation

Relation transitive tab hota hai jab:

Agar (a,b) ∈…