Inverse Relation \( R^{-1} \)

📺 Video Explanation

📝 Question

Let:

\[ A = \{2,3,4\}, \quad B = \{1,3,7\} \]

Relation \( R \) is defined as:

\[ R = \{(x,y) : x \in A,\ y \in B,\ x < y\} \]

Find \( R^{-1} \).

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✅ Solution

🔹 Step 1: Find Relation \( R \)

Check all pairs where \( x < y \):

• For \( x = 2 \): \( (2,3), (2,7) \)
• For \( x = 3 \): \( (3,7) \)
• For \( x = 4 \): \( (4,7) \)

So,

\[ R = \{(2,3), (2,7), (3,7), (4,7)\} \]

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🔹 Step 2: Find Inverse Relation

Inverse relation is obtained by swapping each pair:

\[ R^{-1} = \{(y,x) : (x,y) \in R\} \]

So:

\[ (3,2), (7,2), (7,3), (7,4) \]

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🎯 Final Answer

\[ \boxed{R^{-1} = \{(3,2), (7,2), (7,3), (7,4)\}} \]

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🚀 Exam Insight

• First list all pairs correctly
• Inverse = swap coordinates
• Domain and range get interchanged