Relation \( |x-y| \leq 1 \) on \( \mathbb{Z} \)
📺 Video Explanation
📝 Question
Let \( R \) be a relation on the set of integers \( \mathbb{Z} \) defined by:
\[ (x,y)\in R \iff |x-y|\leq 1 \]
Then, \( R \) is:
- (a) reflexive and transitive
- (b) reflexive and symmetric
- (c) symmetric and transitive
- (d) an equivalence relation
✅ Solution
To check the relation, test reflexive, symmetric, and transitive properties.
🔹 Reflexive
For every integer \( x \),
\[ |x-x| = 0 \leq 1 \]
✔ Reflexive.
🔹 Symmetric
If \[ |x-y|\leq 1 \] then \[ |y-x| = |x-y|\leq 1 \]
✔ Symmetric.
🔹 Transitive
Check a counterexample:
Take: \[ x=1,\quad y=2,\quad z=3 \]
Then:
\[ |1-2|=1\leq1 \] and \[ |2-3|=1\leq1 \]
So, \[ (1,2)\in R,\quad (2,3)\in R \]
But:
\[ |1-3|=2>1 \]
Thus, \[ (1,3)\notin R \]
❌ Not transitive.
🎯 Final Answer
\[ \boxed{\text{R is reflexive and symmetric}} \]
✔ Correct option: (b)
🚀 Exam Shortcut
- Absolute value relations are often symmetric
- Always test transitivity with nearby integers
- One counterexample is enough to reject transitivity