Question

Let A be the set of first natural numbers and let R be a relation on A defined as follows :

$(x,y)ϵR\iff x\le y$

Express R and $R^{-1}$ as sets of ordered pairs. Determine also

(i) The domain of $R^{-1}$

(ii) The range of R.

Answer 1

We have,

A = {1, 2, 3, 4, 5}

[$\because$ A is the set of first five natrual number :]

It is given that R be a relation on A defined as $(x,y)ϵR\iff x\le y$

For the elements of the given sets A and A, we find that

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, and 5 = 5

$\therefore (1,1)ϵR,(1,2)ϵR,(1,3)ϵR,(1,4)ϵR,(1,5)ϵR,(2,2)ϵR,(2,3)ϵR,(2,4)ϵR,(2,5)ϵR,(3,3)ϵR,(3,4)ϵR,(3,5)ϵR,(4,4)ϵR,(4,5)ϵR,$ and $(5,5)ϵR,$

Thus,

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)}

Also,

$R^{-1}=$ {(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),}

(i) Domain $\left(R^{-1}\right)$ = (1, 2, 3, 4, 5)

(ii) Range (R) = (1, 2, 3, 4, 5}