Relation \( x + 4y = 10 \) on \( \mathbb{N} \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{N} \) be defined as:
\[ (x, y) \in R \iff x + 4y = 10 \]
Check whether \( R \) is reflexive, symmetric, and transitive.
✅ Solution
🔹 Step 1: Find Possible Pairs
\[ x + 4y = 10 \Rightarrow x = 10 – 4y \]
For \( x,y \in \mathbb{N} \):
- \( y = 1 \Rightarrow x = 6 \Rightarrow (6,1) \)
- \( y = 2 \Rightarrow x = 2 \Rightarrow (2,2) \)
- \( y = 3 \Rightarrow x = -2 \) (not in \( \mathbb{N} \))
So, \[ R = \{(6,1),(2,2)\} \]
🔹 Step 2: Reflexive
Reflexive requires: \[ (x,x) \in R \quad \forall x \in \mathbb{N} \]
Only \( (2,2) \) is present, not all \( (x,x) \)
❌ Therefore, the relation is Not Reflexive.
🔹 Step 3: Symmetric
Check: \[ (6,1) \in R \]
But: \[ (1,6) \notin R \ (\text{since } 1 + 24 \neq 10) \]
❌ Therefore, the relation is Not Symmetric.
🔹 Step 4: Transitive
Check chains:
\[ (6,1) \text{ and } (1, ?) \notin R \]
\[ (2,2),(2,2) \Rightarrow (2,2) \in R \]
No violating chains exist.
✔ Therefore, the relation is Transitive.
🎯 Final Answer
✔ Reflexive: No
✔ Symmetric: No
✔ Transitive: Yes
\[ \therefore R \text{ is transitive only} \]
🚀 Exam Insight
- First find valid ordered pairs
- Small relations → often transitive (vacuous cases)
- Check symmetry using reverse pairs