Relations & Functions – Basics + Types of Functions | JEE Main Complete Guide

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Relations & Functions Kya Hote Hain?

Relations & Functions Mathematics ka ek fundamental topic hai jo Sets ke concepts par based hota hai. Is chapter ko samajhne ke liye sabse pehle ordered pair, Cartesian product aur relation ko clearly understand karna zaroori hai.

Function ko simple language me ek rule ki tarah samajh sakte hain jo har input ko exactly ek output assign karta hai.

Is topic me JEE Main ke questions generally relation ki definition, domain-range, function test aur functions ke different types par based hote hain.

Is guide me hum step by step dekhenge:

• Ordered Pair
• Cartesian Product
• Relation
• Domain and Range
• Function
• Domain and Codomain
• One-One Function
• Many-One Function
• Into Function
• Onto Function
• Bijective Function

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Ordered Pair Kya Hota Hai?

Ordered pair do elements ka ek pair hota hai jise generally is form me represent kiya jata hai:

(a, b)

Yahan:

• a = First element
• b = Second element

Ordered pair me order important hota hai.

For example:

(1, 2) ≠ (2, 1)

Kyuki first ordered pair me first element 1 hai aur second element 2 hai, jabki second pair me first element 2 hai aur second element 1 hai.

Ye Sets se different important point hai. Set me order generally matter nahi karta, lekin ordered pair me order matter karta hai.

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Cartesian Product

Do sets A aur B diye gaye hon, to unka Cartesian Product A × B un sabhi ordered pairs ka set hota hai jisme first element A se aur second element B se belong karta hai.

Definition:

A × B = {(a,b) : a ∈ A, b ∈ B}

Example:

A = {1,2}

B = {3,4}

Then:

A × B = {(1,3), (1,4), (2,3), (2,4)}

Notice karein ki first position ke elements A se aaye hain aur second position ke elements B se.

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Cartesian Product Ki Cardinality

Agar:

n(A) = m

and:

n(B) = n

then:

n(A × B) = n(A) × n(B)

Example:

A me 2 elements hain aur B me 3 elements hain.

Therefore:

n(A × B) = 2 × 3 = 6

Yani A × B me total 6 ordered pairs honge.

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Relation Kya Hota Hai?

Ek relation R from A → B, Cartesian product A × B ka subset hota hai.

Mathematically:

R ⊆ A × B

Example:

A = {1,2}

B = {3,4}

Cartesian product:

A × B = {(1,3),(1,4),(2,3),(2,4)}

Agar:

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

to R, A × B ka subset hai.

Therefore:

R is a relation from A → B.

Simple rule:

Relation → Cartesian Product ka subset

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Domain of a Relation

Kisi relation ke ordered pairs me jo first elements hote hain, unka set relation ka Domain kehlata hai.

Example:

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

First elements:

1, 2, 5

Therefore:

Domain = {1,2,5}

Shortcut:

Domain → First Elements → Inputs

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Range of a Relation

Kisi relation ke ordered pairs me jo second elements actually appear karte hain, unka set relation ka Range kehlata hai.

For:

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

Second elements:

3, 4, 4

Set me duplicate elements repeat nahi hote, therefore:

Range = {3,4}

Shortcut:

Range → Actual Outputs

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Function Kya Hota Hai?

Function ek special type ka relation hai.

A relation:

f : A → B

function tab hota hai jab A ke every element ka B me exactly one image ho.

Simple language:

1 Input → Exactly 1 Output

Ye function ka sabse important rule hai.

Agar ek input ke do different outputs hain, to wo function nahi hoga.

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Function Test

Allowed Case

Suppose mapping hai:

1 → 3

2 → 4

3 → 4

Yahan:

• 1 ka output 3 hai
• 2 ka output 4 hai
• 3 ka output 4 hai

Har input ka exactly one output hai.

Therefore:

It is a function.

Important: Different inputs ka same output hona allowed hai.

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Not Allowed Case

Suppose:

1 → 3

1 → 4

Yahan same input 1 ke do different outputs hain.

Therefore:

It is NOT a function.

Golden rule:

Same input → Two outputs ❌

Different inputs → Same output ✅

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Domain, Codomain and Range

Agar function hai:

f : A → B

to:

A → Domain

B → Codomain

f(A) → Range

Yahan Domain wo set hai jahan se inputs liye ja rahe hain.

Codomain wo target set hai jisme outputs belong karte hain.

Range sirf un outputs ka set hai jo actually function ke through obtain hote hain.

Isliye important relation:

Range ⊆ Codomain

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Codomain vs Range

Ye JEE Main me ek common conceptual trap hai.

Suppose:

f : A → B

and B me elements hain:

B = {a,b,c,d}

Agar function ke outputs sirf:

{a,b,c}

hain, to:

Codomain = {a,b,c,d}

Range = {a,b,c}

Therefore:

Range ⊂ Codomain

Is case me function Into hoga.

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One-One Function / Injective Function

A function ko One-One ya Injective kaha jata hai jab different inputs ke outputs bhi different hon.

Simple rule:

Different Inputs → Different Outputs

Mathematical condition:

f(a) = f(b) ⇒ a = b

Example:

1 → 2

2 → 4

3 → 6

Yahan kisi bhi two different inputs ka output same nahi hai.

Therefore function One-One hai.

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Many-One Function

Agar different inputs ka same output ho sakta hai, to function Many-One ho sakta hai.

Condition:

f(a) = f(b)

but:

a ≠ b

Example:

1 → 4

2 → 4

3 → 6

Yahan 1 aur 2 different inputs hain, lekin dono ka output 4 hai.

Therefore function Many-One hai.

Important: Many-One hona function ke against nahi hai. Different inputs ka same output allowed hai.

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Into Function

Agar function ka range codomain se chhota hai, yani codomain ke kuch elements ka koi pre-image nahi hai, to function ko Into Function kehte hain.

Condition:

Range ⊂ Codomain

Example:

A = {1,2,3}

B = {a,b,c,d}

Suppose:

1 → a

2 → b

3 → c

Then:

Range = {a,b,c}

Codomain = {a,b,c,d}

Element d ka koi pre-image nahi hai.

Therefore:

Range ⊂ Codomain

So function is:

Into

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Onto Function / Surjective Function

A function ko Onto ya Surjective kaha jata hai jab codomain ka har element actually output ke form me appear kare.

Condition:

Range = Codomain

Matlab codomain ka koi bhi element unused nahi hona chahiye.

Example:

A = {1,2,3}

B = {a,b,c}

Suppose:

1 → a

2 → b

3 → c

Then:

Range = {a,b,c}

Codomain = {a,b,c}

Therefore:

Range = Codomain

Hence function is:

Onto / Surjective

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Bijective Function

A function jo simultaneously One-One + Onto ho, use Bijective Function kaha jata hai.

Therefore:

Injective + Surjective = Bijective

Bijective function me:

• Different inputs ke different outputs hote hain.
• Codomain ka har element use hota hai.
• Har output ka exactly one pre-image hota hai.

Simple way:

Every output has exactly one pre-image.

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One-One vs Many-One

Type	Condition	Example Idea
One-One	Different inputs → Different outputs	1 → a, 2 → b
Many-One	Different inputs → Same output allowed	1 → a, 2 → a

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Into vs Onto

Type	Condition
Into	Range ⊂ Codomain
Onto	Range = Codomain

Memory trick:

Into → Some codomain elements unused

Onto → All codomain elements used

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Quick Comparison of Function Types

Type	Key Condition
One-One	Different input → Different output
Many-One	Different inputs → Same output allowed
Into	Range ⊂ Codomain
Onto	Range = Codomain
Bijective	One-One + Onto

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Important Trap: One-One Does Not Mean Onto

One-One aur Onto completely different properties hain.

Ek function:

One-One + Into

ho sakta hai.

Similarly:

Many-One + Onto

bhi possible hai.

And:

Many-One + Into

bhi possible hai.

Sirf jab function:

One-One + Onto

ho, tab wo Bijective hota hai.

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JEE Main Solved Question

Question: Given:

f : A → B

where:

A = {1,2,3}

B = {a,b,c}

and:

f = {(1,a),(2,b),(3,c)}

Determine the type of function.

Step 1: Check One-One

Outputs hain:

a, b, c

Teeno outputs different hain.

Therefore:

f is One-One.

Step 2: Check Onto

Codomain:

B = {a,b,c}

Range:

{a,b,c}

Therefore:

Range = Codomain

So function is Onto.

Final Answer

f = One-One + Onto

Therefore:

f is Bijective.

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Another JEE Main Example

Question: Let:

f : A → B

where:

A = {1,2,3}

B = {a,b,c,d}

and:

f(1) = a, f(2) = b, f(3) = c

Determine whether the function is Into or Onto.

Solution

Range:

{a,b,c}

Codomain:

{a,b,c,d}

Since:

Range ⊂ Codomain

element d ka koi pre-image nahi hai.

Therefore:

Function is Into.

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Function Identify Karne Ka Fast Method

JEE Main me mapping ya ordered pairs diye ho to ye sequence follow karein:

1. Check every input.
2. Har input ka exactly one output hai?
3. Agar nahi, to function nahi hai.
4. Agar haan, to outputs compare karein.
5. Different inputs ke outputs repeat ho rahe hain to Many-One.
6. Outputs repeat nahi ho rahe to One-One.
7. Range aur Codomain compare karein.
8. Range = Codomain → Onto.
9. Range ⊂ Codomain → Into.
10. One-One + Onto → Bijective.

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Common Traps in Relations & Functions

• Ordered pair me order ignore karna: (1,2) aur (2,1) same nahi hain.
• Same input ke two outputs ko function maan lena: Ye function nahi hota.
• Different inputs ke same output ko invalid samajhna: Ye allowed hai aur Many-One function ho sakta hai.
• Codomain ko Range samajhna: Range sirf actual outputs ka set hai.
• One-One ko Onto samajhna: Dono separate conditions hain.
• Raw output count se Onto decide karna: Range ko actual codomain ke saath compare karna hota hai.
• Relation aur Function ko same samajhna: Every function is a relation, but every relation is not necessarily a function.

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Relation vs Function

Relation	Function
R ⊆ A × B	Special type of relation
Input ke multiple outputs possible ho sakte hain	Every input has exactly one output
General mapping	Strict input-output rule

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JEE Main Exam Perspective

Relations & Functions JEE Main Mathematics ka important chapter hai. Is topic ke basics clear hone par functions ke advanced concepts ko samajhna kaafi easy ho jata hai.

JEE Main ke liye especially in concepts ko strong karein:

• Ordered Pair
• Cartesian Product
• Relation as a subset of A × B
• Domain
• Range
• Codomain
• Function definition
• One-One / Injective
• Many-One
• Into
• Onto / Surjective
• Bijective

Mapping diagram ya ordered pairs ka question aaye to sabse pehle function test perform karein. Uske baad One-One aur Onto conditions separately check karein.

Ek function ko Bijective declare karne ke liye dono conditions satisfy honi chahiye:

One-One + Onto = Bijective

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Important Formula & Concept Sheet

Cartesian Product:

A × B = {(a,b) : a ∈ A, b ∈ B}

Number of Ordered Pairs:

n(A × B) = n(A)n(B)

Relation:

R ⊆ A × B

Domain:

First Elements / Inputs

Range:

Actual Outputs

Function:

Every input → Exactly One Output

One-One:

f(a) = f(b) ⇒ a = b

Into:

Range ⊂ Codomain

Onto:

Range = Codomain

Bijective:

One-One + Onto

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Quick Revision Box

Relation → R ⊆ A × B

Function → 1 Input → Exactly 1 Output

Domain → Inputs

Codomain → Target Set

Range → Actual Outputs

One-One → No Output Repeats

Many-One → Output Can Repeat

Into → Range ⊂ Codomain

Onto → Range = Codomain

Bijective → One-One + Onto

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Practice Questions

1. If A = {1,2} and B = {3,4,5}, find A × B and n(A × B).
2. For R = {(1,3),(2,4),(5,4)}, find the domain and range.
3. Determine whether {(1,2),(2,3),(3,4)} represents a function.
4. Determine whether {(1,2),(1,3),(2,4)} represents a function.
5. If f : A → B is One-One and Into, what can be said about its range and codomain?
6. If Range = Codomain and the function is One-One, identify the type of function.
7. For f : A → B, if two different elements of A have the same image in B, what type of function can it be?
8. If Range is a proper subset of Codomain, identify whether the function is Into or Onto.

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Frequently Asked Questions (FAQs)

Relation kya hota hai?

A relation R from A to B, Cartesian product A × B ka subset hota hai. Mathematically, R ⊆ A × B.

Function kya hota hai?

Function ek special relation hai jisme domain ke har element ka codomain me exactly one image hota hai.

Kya different inputs ka same output ho sakta hai?

Haan. Different inputs ka same output allowed hai. Aisa function Many-One ho sakta hai.

Kya same input ke two outputs ho sakte hain?

Nahi. Agar same input ke two different outputs hain, to given relation function nahi hoga.

Domain aur Range me difference kya hai?

Domain ordered pairs ke first elements ka set hota hai, jabki Range actual second elements ya outputs ka set hota hai.

Codomain aur Range same hote hain?

Always nahi. Range, Codomain ka subset hota hai. Agar Range = Codomain ho, to function Onto hota hai.

One-One function kya hota hai?

One-One function me different inputs ke different outputs hote hain. Isse Injective function bhi kehte hain.

Onto function kya hota hai?

Onto function me codomain ka har element actual output ke form me appear karta hai. Therefore Range = Codomain.

Bijective function kya hota hai?

Jo function simultaneously One-One aur Onto ho, use Bijective Function kehte hain.

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Download PDF Notes

JEE Main Relations & Functions ke revision notes ko yahan directly read karein:

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Final Thoughts

Relations & Functions ko master karne ke liye sabse pehle Sets aur Ordered Pairs ke concepts clear hone chahiye. Ordered pair me order important hota hai, jabki Cartesian Product do sets se possible ordered pairs create karta hai.

Relation ko simple rule se yaad rakhein:

Relation → R ⊆ A × B

Function relation ka special case hai jisme domain ke har element ka exactly one output hota hai. Isliye function check karte waqt sabse pehle ye dekhein ki koi input multiple outputs to nahi de raha.

Different inputs ka same output hona allowed hai, isliye aisa function Many-One ho sakta hai. Lekin same input ke two different outputs hone par relation function nahi hota.

Function ke types ko yaad rakhne ke liye do separate questions poochiye:

Question 1: Kya output repeat ho raha hai?

Output repeat nahi → One-One

Output repeat ho sakta hai → Many-One

Question 2: Kya codomain ka har element use hua?

Range = Codomain → Onto

Range ⊂ Codomain → Into

Finally:

One-One + Onto = Bijective

JEE Main ke questions me agar aap Domain, Codomain aur Range ka difference clearly samajh lete hain aur One-One तथा Onto ko separately test karte hain, to functions ke basic classification questions kaafi easily solve kiye ja sakte hain.

Golden Rule: Function = 1 Input → Exactly 1 Output; Bijective = One-One + Onto.