Question

Let A be the set of first ten natural numbers and let R be relation on A defined by (x, y)$\in R^{\prime }$ x = 2y = 10, i.e. R = {(x, y): $x\in A,y\in A$ and x + 2y = 10}. Express R and $R^{-1}$ as sets of ordered pairs. Determine also (i) domains of R and $R^{-1}$ (ii) ranges of R and $R^{-1}$.

Answer 1

$y=\frac{10-x}{2}$
where both x and y are natural numbers between 1 and 10.
Clearly for x= 2, 4, 6, 8 we get values of xy as 4, 3, 2, 1.
$\therefore$ R = ( 2, 4), (4, 3), (6, 2) (8, 1)}
$\therefore R^{-1}$= {(4, 2) , (3, 4),(2, 6), (1, 8)}
Domain of R is {2, 4, 6, 8}= Range of $R^{{}_1}$
Domain of $R^{-1}$ is { 4, 3, 2, 1} = Range of R