If $R$ is a binary relation on a set $A$ define $R^{-1}$ on $A$.
Let $R=\{(a, b): a, b \in W$ and $3 a+2 b=15\}$ and $3 a+2 b=15\}$, where $W$ is the set of whole numbers.
Express $R$ and $R^{-1}$ as sets of ordered pairs.
Show that (i) dom $(R)=$ range $\left(R^{-1}\right)$ (ii) range $(R)=$ dom $\left(R^{-1}\right)$
3a + 2b = 15
$b=\frac{15-3 a}{2}$
$a=1$ è $b=6$
$a=3$ è $b=3$
$a=5$ è $b=0$
$R=\{(1,6),(3,3),(5,0)\}$
$R^{-1}=\{(6,1),(3,3),(0,5)\}$
The domain of R is the set of first co-ordinates of R
Dom(R) = {1, 3, 5}
The range of R is the set of second co-ordinates of R
Range(R) = {6, 3, 0}
The domain of $R^{-1}$ is the set of first co-ordinates of $R^{-1}$
$\operatorname{Dom}\left(R^{-1}\right)=\{6,3,0\}$
The range of $R^{-1}$ is the set of second co-ordinates of $R^{-1}$
Range $\left(R^{-1}\right)=\{1,3,5\}$
Thus,
$\operatorname{dom}(\mathrm{R})=\operatorname{range}\left(\mathrm{R}^{-1}\right)$
range $(\mathrm{R})=\operatorname{dom}\left(\mathrm{R}^{-1}\right)$