Domain of Relation \( x^2 + y^2 \le 4 \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{Z} \) be defined as:
\[ (x,y) \in R \iff x^2 + y^2 \le 4 \]
Find the domain of \( R \).
✅ Solution
🔹 Step 1: Understand Domain
Domain = set of all values of \( x \) such that there exists \( y \in \mathbb{Z} \) satisfying:
\[ x^2 + y^2 \le 4 \]
🔹 Step 2: Possible Values of \( x \)
Since: \[ x^2 \le 4 \]
So: \[ x = -2, -1, 0, 1, 2 \]
🔹 Step 3: Verify Each Value
- \( x = \pm 2 \Rightarrow x^2 = 4 \Rightarrow y^2 \le 0 \Rightarrow y = 0 \) ✔
- \( x = \pm 1 \Rightarrow x^2 = 1 \Rightarrow y^2 \le 3 \Rightarrow y = -1,0,1 \) ✔
- \( x = 0 \Rightarrow x^2 = 0 \Rightarrow y^2 \le 4 \Rightarrow y = -2,-1,0,1,2 \) ✔
Each value of \( x \) has at least one integer \( y \).
🔹 Step 4: Write Domain
\[ \text{Domain} = \{-2, -1, 0, 1, 2\} \]
🎯 Final Answer
\[ \boxed{\{-2, -1, 0, 1, 2\}} \]
🚀 Exam Insight
- Think: \( x^2 \le 4 \) ⇒ limits x directly
- Check at least one y exists
- Represents integer points inside a circle of radius 2