JEE Main Set Theory: Identify Relation Properties in 60 Sec 💡

 

❓ Concept

🎬 Reflexive, Transitive but NOT Symmetric — Easy JEE Trick in 60 Sec

Relations ke chapter mein yeh 3 words sabse zyada confusion create karte hain:
Reflexive, Transitive, Symmetric.

Good news?
👉 Inko samajhne ke liye formula nahi, sirf arrow-logic chahiye 🔥

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🖼️ Concept Image

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✍️ Short Explanation

JEE mein relations ko ordered pairs se zyada,
arrows on elements ke jaise socho.

Jaise hi arrows clear hue,
✔ property identify
✔ options reject
✔ counting easy

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🔹 Step 1 — Reflexive Relation Rule (NON-NEGOTIABLE ⚠️)**

Agar set:

$$A=\{1,2,3\}$$

Toh reflexive hone ke liye:

$$(1,1),(2,2),(3,3)$$

MUST be present.

📌 Inme se ek bhi missing hua
→ relation reflexive hi nahi rahega.

Self-loops = compulsory.

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🔹 Step 2 — Transitive Relation Rule (CHAIN EFFECT 🔥)**

Rule:

$$(a,b)\in R \ \text{and}\ (b,c)\in R \Rightarrow (a,c)\in R$$

Arrow language:

• Agar a → b

• Aur b → c

👉 Toh a → c compulsory.

📌 Jab bhi arrow-chain dikhe,
third arrow automatically add ho jaata hai.

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🔹 Step 3 — What does NOT Symmetric mean?

Symmetric ka matlab hota hai:

$$(a,b)\in R\Rightarrow (b,a)\in R$$

So NOT symmetric ka matlab:
👉 At least ONE exception hona chahiye.

Example:

$$(1,2)\in R\text{but}(2,1)\mathrm{\notin }R$$

✔️ Bas itna kaafi hai relation ko NOT symmetric banane ke liye.

📌 Zaroori nahi ki sab pairs asymmetric hon —
ek bhi chal jaata hai.

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🔹 Step 4 — What the Question is REALLY Asking

Typical JEE question ka hidden meaning hota hai:

Make a relation that:

• ✔ Reflexive (all self-loops included)

• ✔ Transitive (arrow chains complete)

• ❌ NOT symmetric (at least one one-way arrow)

• ✔ Includes a specific pair (like $(1,2)$)

• ✔ Total pairs ≤ given limit (e.g. ≤ 6)

👉 Phir possible relations count karne hote hain.
(Concept samajh lo — counting apne aap easy ho jaati hai 😎)

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⭐ SUPER SHORT SUMMARY (WRITE IN A BOX)

Reflexive → All $(a,a)$ MUST be there
Transitive → If $a\to b$ & $b\to c$, then $a\to c$
NOT Symmetric → At least one $a\to b$ without $b\to a$

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✅ Final Takeaway

🧠 Relation Master Rule

• Self-loops = reflexive

• Arrow chains = transitive

• One-way arrow = not symmetric

Agar arrows samajh aa gaye,
👉 Relations ka aadha chapter khatam 😄