Range of Relation \( x + 2y = 8 \)

📺 Video Explanation

📝 Question

Let relation \( R \) on \( \mathbb{N} \) be defined as:

\[ (x,y) \in R \iff x + 2y = 8 \]

Find the range of \( R \).

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✅ Solution

🔹 Step 1: Express x in terms of y

\[ x + 2y = 8 \Rightarrow x = 8 – 2y \]

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🔹 Step 2: Find possible values of y

Since \( x, y \in \mathbb{N} \), we need \( x > 0 \).

\[ 8 – 2y > 0 \Rightarrow 2y < 8 \Rightarrow y < 4 \]

So possible values: \[ y = 1, 2, 3 \]

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🔹 Step 3: Find corresponding x

• \( y = 1 \Rightarrow x = 6 \)
• \( y = 2 \Rightarrow x = 4 \)
• \( y = 3 \Rightarrow x = 2 \)

So ordered pairs: \[ (6,1), (4,2), (2,3) \]

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🔹 Step 4: Write Range

Range = set of all second elements \( y \):

\[ \{1,2,3\} \]

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🎯 Final Answer

\[ \boxed{\{1,2,3\}} \]

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🚀 Exam Insight

• Range = second elements only
• Ensure values satisfy natural number condition
• Always check positivity