Class 12 Mathematics: Relations aur Functions - Notes

Introduction

Is chapter mein hum sets ke beech relation (sambandh) aur mapping (phalane) ke baare mein padhenge. Ye chapter Class 12 Maths ka pehla chapter hai aur calculus, algebra aur real-life applications ke liye foundation provide karta hai. Yahan hum different types of relations, functions, aur unke properties ko examples ke saath samjhenge.

Topics Covered

• Types of relations - empty, universal, reflexive, symmetric, transitive, equivalence
• Functions - definition, domain, codomain, range
• Types of functions - injective (one-one), surjective (onto), bijective
• Composition of functions aur invertible functions

Here we have provided NCERT notes for Class 12 Mathematics in hindi Language, Just select the chapters below to get notes of the same:

समुच्चय सिद्धांत तथा संबंध

लघुगणक, घातांक, एव करणी, आंशिक भिन्ने

समीक्ष संख्याये

श्रेणियाँ

वर्ग समीकरण एवं असमिकाये

क्रमचय एवं संचय

दुपद प्रमेय तथा गणितीय आगमन

चरघातांकी तथा लघुगणकीय श्रेणियाँ

सारणिक तथा आव्यूह

त्रिकोणमितीय फलन एवं त्रिकोणमितीय अनुपात

त्रिकोणमितीय समीकरण एवं असमिकाये त्रिभुजो के गुण, ऊँचाई एवं दूरी

प्रतिलोम त्रिकोणमितीय फलन

अतिपरवलयिक फलन

समकोणीय कार्तीय निर्देशांक

सरल रेखा

रेखायुग्म

वृत्त तथा वृत्तों का निकाय

शांकव परिच्छेद

सदिश बीजगणित

त्रिविमीय निर्देशांक ज्यामिति

फलन, सीमा, सातंत्य तथा अवकलनीयता

अवकलन तथा अवकलन के अनुप्रयोग

अनिश्चित समाकलन

निश्चित समाकलन एवं वक्रों से घिरा क्षेत्रफल

अवकल समीकरण

स्थिति विज्ञान

गति विज्ञान

प्रायिकता

केन्द्रीय प्रवृत्ति के माप

सहसम्बन्ध एवं समाश्रयण

आंकिक विधियाँ

रैखिक प्रक्रमन

गणितीय तर्कशास्त्र एवं बूलीय बीजगणित

संगणना तथा द्विआधारी संक्रियायें

1. Relation (Sambandh) Kya Hai?

Do sets A aur B ke beech ek relation R, A × B ka ek subset hota hai. Yaani, ordered pairs (a, b) jahan a ∈ A aur b ∈ B. Agar (a, b) ∈ R, to hum likhte hain a R b.

Representation of a Relation

• Roster form: R = {(1,2), (2,3), ...}
• Set-builder form: R = {(x, y) : x ∈ A, y ∈ B, aur condition}

Types of Relations (Sambandh ke Prakar)

• Empty Relation: R mein koi ordered pair nahi hota.
• Universal Relation: R = A × B (sare possible pairs).
• Reflexive Relation: Har a ∈ A ke liye (a, a) ∈ R.
• Symmetric Relation: Agar (a, b) ∈ R, to (b, a) ∈ R.
• Transitive Relation: Agar (a, b) ∈ R aur (b, c) ∈ R, to (a, c) ∈ R.
• Equivalence Relation: Relation jo reflexive, symmetric aur transitive ho.

2. Functions (Phalane) Kya Hai?

Set A se set B mein ek function f ek aisa relation hota hai jismein A ke har element ka exactly ek hi image B mein hota hai. Write f: A → B.

Domain, Codomain, Range

• Domain: Set A (jahan se input aata hai)
• Codomain: Set B (jahan output jaa sakta hai)
• Range: Actual set of outputs f(A) = {f(a) : a ∈ A}

Types of Functions

• Injective (One-One): Different inputs ke different outputs hote hain. f(a1) = f(a2) implies a1 = a2.
• Surjective (Onto): Codomain ka har element kisi na kisi input ka image hota hai. Range = Codomain.
• Bijective: Jo injective bhi ho aur surjective bhi.

3. Composition of Functions aur Invertible Functions

Agar f: A → B aur g: B → C do functions hain, to unka composition gof: A → C is tarah define hota hai: (gof)(x) = g(f(x)).

Ek function f invertible hota hai agar uska inverse function f⁻¹ exist kare. Iske liye f ko bijective hona zaroori hai. Inverse function ko f⁻¹(y) = x, jahan f(x) = y.

4. Important Examples

Example 1: Check karo ki kya relation R = {(1,1), (2,2), (3,3), (1,2), (2,1)} set {1,2,3} par reflexive, symmetric, transitive hai?
Solution: Reflexive? Haan, kyunki (1,1), (2,2), (3,3) hain. Symmetric? Haan, (1,2) aur (2,1) dono hain. Transitive? Haan, check karo: (1,2) aur (2,1) se (1,1) milta hai jo already hai. To ye ek equivalence relation hai.

Example 2: f: R → R, f(x) = 2x + 3. Kya f invertible hai?
Solution: f injective hai kyunki agar 2x+3 = 2y+3 to x=y. Surjective hai kyunki kisi bhi y ke liye x = (y-3)/2 milta hai. To bijective hai, isliye invertible. Inverse f⁻¹(y) = (y-3)/2.

5. Quick Revision Tips

• Relations ko visualize karne ke liye arrow diagram aur matrix ka upyog karein.
• Function ki injectivity check karne ke liye horizontal line test istemal karein (if graph available).
• Inverse function dhundhne ke liye ‘swap x and y’ method yaad rakhein.
• CBSE board exams mein relation aur function ke proofs par focus hota hai.