Find R⁻¹ if R is Defined by y = x − 3
Question
If \( R \) is a relation from set
\[ A=\{11,12,13\} \]
to set
\[ B=\{8,10,12\} \]
defined by
\[ y=x-3, \]
then write \( R^{-1} \).
Solution
Given,
\[ y=x-3 \]
For \( x=11 \),
\[ y=11-3=8 \]
\[ (11,8)\in R \]
For \( x=12 \),
\[ y=12-3=9 \notin B \]
For \( x=13 \),
\[ y=13-3=10 \]
\[ (13,10)\in R \]
Therefore,
\[ R=\{(11,8),(13,10)\} \]
Hence,
\[ R^{-1}=\{(8,11),(10,13)\} \]
\[ \boxed{\{(8,11),(10,13)\}} \]