Class 12 Mathematics: Relations aur Functions - Notes

Introduction

Is chapter mein hum sets ke beech ke sambandhon aur mappings—yaani functions—ko samajhne wale hain. Yeh Class 12 Maths ka pehla chapter hai. Aur iska khel bilkul seedha hai: calculus, algebra, aur rozmarra ki zindagi ke liye solid base. Hum yahan alag-alag tarah ke relations aur functions dekhenge—reflexive se lekar transitive tak—aur unki properties ko examples ke saath aise samjhenge ki concept ka pakka ho jaye, bina kisi jhanjhat ke.

Topics Covered

• Here’s the rewritten paragraph, keeping the heading "Topics Covered" in mind and following all your rules: --- We kick things off with the different flavors of relations. You’ll get a clear breakdown of the empty and universal ones, then move right into reflexive, symmetric. Transitive. And of course, we tie it all together with equivalence relations—the real glue that holds everything together. It’s a packed list, but we take it step by step. --- Plain text, no markdown, all facts intact.
• Functions. What they're, how they work, and the three key pieces that make them tick: domain, codomain, and range. We break down the definition, then get into what actually goes in, where it maps to, and what really comes out the other side.
• Under the hood, this one’s all about mapping. We break down functions into three distinct flavors: injective, where every output gets hit by at most one input—that’s your one-to-one deal. Then surjective, or onto, meaning every possible output is actually used. And finally, bijective, which is the sweet spot where both conditions hold at once. It’s a clean little trio, and once you see the pattern, it sticks.
• Composition of functions and invertible functions—that’s the real meat here. We’re talking about how you chain two functions together, then flip the whole thing around to undo it. If you’ve got a function that’s one-to-one and onto, you can reverse it, and that inverse basically undoes whatever the original did. It’s neat, kind of like a mathematical undo button. So, expect to see how these two ideas link up, why invertibility hinges on composition, and how to spot when a function actually qualifies. That’s the core of what we’re digging into.

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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 relation R ka matlab hota hai ki R, A × B ka ek subset hai. Matlab, ordered pairs (a, b) lete ho, jahan a, A se hai aur b, B se. Ab agar (a, b) is subset mein milta hai, to hum seedha likh dete hain a R b. Bas, itna hi hai.

Representation of a Relation

• Roster form just lists the pairs out, plain and simple—like R = {(1,2), (2,3),...}. That’s it. No fancy setup, no rules to decode. You see exactly what relates to what, one pair at a time. The ellipsis, though, that’s where it gets a little cheeky—it’s basically saying, “you get the pattern, right?” Because if the relation’s huge or infinite, you can’t write every single pair down, obviously. So you jot a few, slap on that “...”. Hope the reader fills in the blanks. It works great for small, tidy relations. For big ones? You’re kind of trusting the dots to do the heavy lifting.
• Set-builder form is just a compact way to write the relation down. You’re basically saying R equals all the ordered pairs (x, y) where x comes from A, y comes from B, and—on top of that—there’s an extra condition tagged on with “aur.” That last part’s the real filter, the thing that actually decides which pairs make the cut. Without it, you’d just have the whole Cartesian product sitting there. But throw in that condition, and suddenly you’ve narrowed things down to exactly the pairs that matter.

Types of Relations (Sambandh ke Prakar)

• Empty relation is pretty much what it sounds like. R simply has no ordered pairs in it at all. Not a single one. Think of it as a completely blank slate—there’s nothing connecting any elements, no links, no pairs, just pure emptiness. That’s the whole deal.
• Universal Relation: here, R equals A × B, which means every single possible pair is included. No filtering, no conditions—just all of it, plain and simple.
• Reflexive Relation: Simply put, for every element a in set A, the pair (a, a) must belong to R. That’s the whole rule—no exceptions. If you’ve got an element in A, it has to be paired with itself. Every single time.
• Symmetric Relation: The idea is dead simple. If (a, b) is sitting inside R, then you’d better believe (b, a) is right there too. No exceptions, no ifs or buts—swap the pair, and it still holds. That’s the whole game.
• Transitive Relation: So here’s the deal. If (a, b) is in R, and (b, c) is also in R, then guess what? (a, c) has to be in R too. That’s the whole rule, plain and simple. You see the pattern—a links to b, b links to c, and boom, a links to c. No shortcuts, no exceptions. It’s like passing a message down a line: if the first person tells the second. The second tells the third, then the first has effectively reached the third, even if they never spoke directly. That’s what makes it transitive. Keep that in mind, and you’ll spot these relations a mile away.
• Equivalence Relation: Ek aisa relation jo teeno cheezein ho — reflexive, symmetric, aur transitive. Matlab, pehle toh har element ka apne aap se rishta hona chahiye, phir agar A ka B se rishta hai toh B ka bhi A se hona zaroori hai, aur aakhir mein agar A ka B se aur B ka C se rishta hai, toh A ka C se bhi hona chahiye. Teeno conditions ek saath poori ho, tabhi use equivalence relation kehte hain.

2. Functions (Phalane) Kya Hai?

Set A se set B mein function f ka matlab hota hai ek aisa relation, jahan A ka har element B mein bilkul ek hi image rakhta hai. Simple si baat hai. Koi element left out nahi, koi element do jagah nahi jaata. Aur hum ise likhte hain f: A → B.

Domain, Codomain, Range

• Domain’s basically set A—that’s where your input comes from. Simple as that.
• Codomain? That’s just set B—the place where outputs are allowed to land. Think of it like a parking lot: everything that comes out of the function has to fit somewhere in B, even if it never actually shows up. The codomain says, "Hey, these are the possible spots," but it doesn’t guarantee every spot gets filled. It’s the theoretical playground for your outputs, not necessarily the ones you’ll see in action. So, yeah, B’s the codomain, plain and simple—where the output can go, not where it always does.
• Look, the range is really the set of outputs you actually get. You define the function, you plug in everything from the domain, and whatever comes out the other side—that collection, and only that collection—is your range. We write it as f(A) = {f(a): a ∈ A}. In plain English, it's all the f(a) values you land on when you run every single a through the function. That's it. No more, no less.

Types of Functions

• Injective, or one-one, functions keep things clean. Give it two different inputs, and you’ll always get two different outputs. No shortcuts, no repeats. Mathematically, that means if f(a1) equals f(a2), then a1 has to be a2. It’s a simple rule, but it holds everything together.
• Surjective (Onto) — har element jo codomain mein hai, wo kisi na kisi input ka image zaroor hota hai. Matlab, koi bhi element left out nahi rehta. Isliye range aur codomain bilkul equal ho jaate hain. Simple si baat hai: jo value tumhare codomain mein hai, wo kahin na kahin se aayi hai.
• Bijective: jab ek function dono kaam karta ho—injective bhi, aur surjective bhi. Yani har element ka ek unique partner ho, aur koi bhi element chhuta na ho. Simple si baat.

3. Composition of Functions aur Invertible Functions

Agar f: A → B aur g: B → C do functions hain, toh unka composition gof: A → C exactly aise samjho. Pehle f lagao, phir jo mile uspar g. Matlab (gof)(x) = g(f(x)). Simple si baat hai, bas order ka khayal rakhna.

Ek function f tab invertible hota hai jab uska inverse, f⁻¹, exist karta hai. Aur yeh tabhi possible hai jab f bijective ho—matlab ekdum perfect pairing. Inverse function ka formula aisa hota hai: f⁻¹(y) = x, jab f(x) = y.

4. Important Examples

Example 1: Let's see if the relation R = {(1,1), (2,2), (3,3), (1,2), (2,1)} on the set {1,2,3} is reflexive, symmetric, and transitive. Solution: Reflexive? Yeah, we've got (1,1), (2,2), and (3,3) all sitting there, so that's a yes. Symmetric? Sure is—if (1,2) is in there, (2,1) is right alongside it. Now transitive? That one needs a quick look. Pair (1,2) with (2,1) and you land on (1,1), which is already present. So everything checks out — this thing's an equivalence relation, plain and simple.

Example 2: f: R → R, f(x) = 2x + 3. Toh kya f invertible hai? Dekhte hain. Pehle check karo injective — agar 2x + 3 = 2y + 3, toh x = y, seedha saaf. Ab surjective — kisi bhi y ke liye, x = (y-3)/2 nikal jata hai, bas. Matlab bijective hai, aur bijective ka matlab invertible. Inverse bhi simple hai: f⁻¹(y) = (y-3)/2.

5. Quick Revision Tips

• Jab aap relations padh rahe ho, toh sirf theory mat padho—arrow diagram aur matrix ka saath le lo. Ye dono cheezein visualize karne mein kaafi madad karti hain, aur revision ke waqt ekdum kaam aati hain. Yaad rahe, jitna zyada visual karte ho, utna zyada dimaag mein baithta hai.
• If the graph's in front of you, the horizontal line test is your best friend for checking injectivity. Just run a line across it—if it touches more than once, the function’s not one-to-one. Simple as that.
• Yeh wala rule kabhi mat bhoolna—inverse function nikalna ho toh bas x aur y ko swap kar do. Simple hai, aur kaam bhi hamesha chal jaata hai.
• Look, if there’s one thing you absolutely can’t ignore while revising for the boards, it’s the proofs in relations and functions. They show up again and again. Not exactly a shock, honestly, since the examiners clearly have a soft spot for them. So, yeah, give those proofs the bulk of your attention. That’s your quick win.