Relation \( x > y \) on \( \mathbb{N} \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{N} \) be defined as:
\[ (x, y) \in R \iff x > y \]
Check whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: Reflexive
A relation is reflexive if: \[ (x, x) \in R \quad \forall x \in \mathbb{N} \]
But: \[ x > x \ \text{is never true} \]
❌ Therefore, the relation is Not Reflexive.
🔹 Step 2: Symmetric
A relation is symmetric if: \[ (x, y) \in R \Rightarrow (y, x) \in R \]
If: \[ x > y \]
Then: \[ y > x \ \text{is false} \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 3: Transitive
A relation is transitive if: \[ (x, y) \in R \text{ and } (y, z) \in R \Rightarrow (x, z) \in R \]
If: \[ x > y \text{ and } y > z \]
Then: \[ x > z \]
✔ Therefore, the relation is Transitive.
🎯 Final Answer
✔ Reflexive: No
✔ Symmetric: No
✔ Transitive: Yes
\[ \therefore R \text{ is transitive only} \]
🚀 Exam Insight
- Strict inequalities (>, <) are never reflexive
- They are never symmetric
- They are always transitive