Question:
Let A be the set of first five natural numbers and let R be a relation on A, defined by $(x, y) \in R \leftrightarrow x \leq y$
Express $\mathbf{R}$ and $\mathbf{R}^{-1}$ as sets of ordered pairs.
Find: dom $\left(R^{-1}\right)$ and range $(R)$.
Solution:
A = {1, 2, 3, 4, 5}
Since, $x \leq y$
$\mathrm{R}=\{(1,1),(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,3),(3,4),(3,5),(4,$, 4), $(4,5),(5,5)\}$
The domain of R is the set of first co-ordinates of R
Dom(R) = {1, 2, 3, 4, 5}
The range of R is the set of second co-ordinates of R
Range(R) = {1, 2, 3, 4, 5}