Domain of Relation \( a^2 + b^2 = 25 \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{Z} \) be defined as:
\[ (a,b) \in R \iff a^2 + b^2 = 25 \]
Find the domain of \( R \).
✅ Solution
🔹 Step 1: Understand Domain
Domain = set of all first elements \( a \) such that there exists \( b \in \mathbb{Z} \) satisfying:
\[ a^2 + b^2 = 25 \]
🔹 Step 2: Find Integer Solutions
We look for integers \( a, b \) such that:
\[ a^2 + b^2 = 25 \]
Possible squares: \[ 25 = 0^2 + 5^2 = 3^2 + 4^2 \]
So, ordered pairs are:
- \( (0, \pm5) \)
- \( (\pm5, 0) \)
- \( (\pm3, \pm4) \)
- \( (\pm4, \pm3) \)
🔹 Step 3: Extract Domain
Domain = all possible values of \( a \):
\[ \{-5, -4, -3, 0, 3, 4, 5\} \]
🎯 Final Answer
\[ \text{Domain} = \{-5, -4, -3, 0, 3, 4, 5\} \]
🚀 Exam Insight
- Recognize Pythagorean triples: (3,4,5)
- Include positive and negative integers
- Domain = first elements only
- Always list all integer solutions