Relation on Polygons Having Same Number of Sides

📺 Video Explanation

📝 Question

Let \( A \) be the set of all polygons. Define relation \( R \) as:

\[ R = \{(P_1, P_2) : P_1 \text{ and } P_2 \text{ have same number of sides}\} \]

Show that \( R \) is an equivalence relation. Also find all elements related to the right-angled triangle \( T \) with sides 3, 4, 5.

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✅ Solution

🔹 Step 1: Reflexive

Every polygon has the same number of sides as itself: \[ P_1 \text{ has same number of sides as } P_1 \]

So, \[ (P_1, P_1) \in R \]

✔ Therefore, the relation is Reflexive. :contentReference[oaicite:0]{index=0}

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🔹 Step 2: Symmetric

If: \[ (P_1, P_2) \in R \]

Then \( P_1 \) and \( P_2 \) have same number of sides.

So, \[ (P_2, P_1) \in R \]

✔ Therefore, the relation is Symmetric. :contentReference[oaicite:1]{index=1}

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🔹 Step 3: Transitive

If: \[ (P_1, P_2) \in R \text{ and } (P_2, P_3) \in R \]

Then all three polygons have the same number of sides.

So, \[ (P_1, P_3) \in R \]

✔ Therefore, the relation is Transitive. :contentReference[oaicite:2]{index=2}

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🎯 Final Conclusion

✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes

\[ \therefore R \text{ is an equivalence relation} \]

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🔹 Elements Related to Triangle \( T(3,4,5) \)

The given triangle \( T \) has: \[ 3 \text{ sides} \]

So, all polygons having 3 sides are related to \( T \).

Hence, the equivalence class is:

\[ [T] = \{\text{all triangles}\} \]

✔ Therefore, all triangles are related to \( T \). :contentReference[oaicite:3]{index=3}

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🚀 Exam Insight

• Relation groups polygons by number of sides
• Each equivalence class = polygons with same sides
• Triangle → class of all triangles
• Quadrilateral → class of all 4-sided polygons