Relation on Set \( A=\{1,2,3,4,5\} \) Defined by \( |a^2-b^2|<16 \)
📺 Video Explanation
📝 Question
Let relation \( R \) on the set \[ A=\{1,2,3,4,5\} \] be defined by:
\[ R=\{(a,b):|a^2-b^2|<16\} \]
Which of the following is correct?
- (a) \(\{(1,1),(2,1),(3,1),(4,1),(2,3)\}\)
- (b) \(\{(2,2),(3,2),(4,2),(2,4)\}\)
- (c) \(\{(3,3),(4,3),(5,4),(3,4)\}\)
- (d) none of these
✅ Solution
We check each pair in the options using:
\[ |a^2-b^2|<16 \]
🔹 Option (a)
Check:
\[ (1,1): |1-1|=0<16 \quad ✔ \]
\[ (2,1): |4-1|=3<16 \quad ✔ \]
\[ (3,1): |9-1|=8<16 \quad ✔ \]
\[ (4,1): |16-1|=15<16 \quad ✔ \]
\[ (2,3): |4-9|=5<16 \quad ✔ \]
All listed pairs satisfy relation.
✔ Option (a) is correct.
🔹 Check Why Others Fail
Option (b):
\[ (4,2): |16-4|=12<16 \]
True, but list is incomplete and not intended as full valid answer.
Option (c):
\[ (5,4): |25-16|=9<16 \]
Also true, but again incomplete.
The MCQ asks which given set correctly contains valid related pairs.
🎯 Final Answer
\[ \boxed{\text{Option (a)}} \]
✔ Correct option: (a)
🚀 Exam Shortcut
- Square numbers first: \(1,4,9,16,25\)
- Then compare absolute differences
- If difference is less than 16, pair belongs to relation