Range of Relation from \( A \) to \( B \) Defined by \( x>y \)

📺 Video Explanation

📝 Question

Let:

\[ A=\{1,2,3\}, \quad B=\{1,4,5,9\} \]

A relation \( R \) from \( A \) to \( B \) is defined by:

\[ xRy \iff x>y \]

Find the range of \( R \).

• (a) \(\{1,4,6,9\}\)
• (b) \(\{4,6,9\}\)
• (c) \(\{1\}\)
• (d) none of these

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✅ Solution

Relation from \( A \) to \( B \) means:

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

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🔹 Check each element of A

For: \[ x=1 \] No element in \( B \) is less than 1.

So no pair.

For: \[ x=2 \] Only: \[ 2>1 \] So: \[ (2,1) \]

For: \[ x=3 \] Only: \[ 3>1 \] So: \[ (3,1) \]

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🔹 Relation Set

\[ R=\{(2,1),(3,1)\} \]

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🔹 Range

Range = set of second components:

\[ \{1\} \]

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🎯 Final Answer

\[ \boxed{\{1\}} \]

✔ Correct option: (c)

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🚀 Exam Shortcut

• Range means second elements of ordered pairs
• Check condition carefully from A to B
• List valid pairs first, then extract range