Relation \(R\) on the First Five Natural Numbers Where \(x\le y\)
Question
Let \[ A=\{1,2,3,4,5\} \] and let \(R\) be a relation on \(A\) defined as:
\[ (x,y)\in R \iff x\le y \]
Express \(R\) and \(R^{-1}\) as sets of ordered pairs. Determine also:
(i) the domain of \(R^{-1}\)
(ii) the range of \(R\)
Solution
\[ R= \{ (1,1),(1,2),(1,3),(1,4),(1,5), \]
\[ (2,2),(2,3),(2,4),(2,5), \]
\[ (3,3),(3,4),(3,5), \]
\[ (4,4),(4,5), (5,5) \} \]
Interchanging the coordinates of each ordered pair,
\[ R^{-1}= \{ (1,1),(2,1),(3,1),(4,1),(5,1), \]
\[ (2,2),(3,2),(4,2),(5,2), \]
\[ (3,3),(4,3),(5,3), \]
\[ (4,4),(5,4), (5,5) \} \]
(i) Domain of \[ R^{-1} \]
\[ \boxed{ \{1,2,3,4,5\} } \]
(ii) Range of \[ R \]
\[ \boxed{ \{1,2,3,4,5\} } \]