Express \(R\) and \(R^{-1}\) as Sets of Ordered Pairs
Question
Let \(R\) be a relation in \(N\) defined by
\[ (x,y)\in R \iff x+2y=8. \]
Express \(R\) and \(R^{-1}\) as sets of ordered pairs.
Solution
Given,
\[ x+2y=8 \]
For \[ y=1,\quad x=6 \]
\[ (6,1) \]
For \[ y=2,\quad x=4 \]
\[ (4,2) \]
For \[ y=3,\quad x=2 \]
\[ (2,3) \]
Therefore,
\[ \boxed{ R=\{(6,1),(4,2),(2,3)\} } \]
Interchanging the coordinates,
\[ R^{-1}=\{(1,6),(2,4),(3,2)\} \]
Therefore,
\[ \boxed{ R^{-1}=\{(1,6),(2,4),(3,2)\} } \]