Relation \( |x^2 – y^2| < 1 \) on \( A = \{1,2,3,4,5\} \)
📺 Video Explanation
📝 Question
Let relation \( R \) on set \( A = \{1,2,3,4,5\} \) be defined as:
\[ (x,y) \in R \iff |x^2 – y^2| < 1 \]
Write \( R \) as a set of ordered pairs.
✅ Solution
🔹 Step 1: Understand Condition
\[ |x^2 – y^2| < 1 \]
Since \( x, y \in \{1,2,3,4,5\} \), their squares are integers.
The only integer with absolute value less than 1 is:
\[ 0 \]
So:
\[ x^2 – y^2 = 0 \Rightarrow x^2 = y^2 \]
🔹 Step 2: Solve Condition
\[ x^2 = y^2 \Rightarrow x = y \quad (\text{since all are positive}) \]
🔹 Step 3: Write Ordered Pairs
All pairs where \( x = y \):
\[ (1,1), (2,2), (3,3), (4,4), (5,5) \]
🎯 Final Answer
\[ R = \{(1,1), (2,2), (3,3), (4,4), (5,5)\} \]
🚀 Exam Insight
- |integer| < 1 ⇒ only 0
- So condition reduces to equality
- Gives identity relation
- Always check domain type (positive integers here)