Question

Let $A=\{2,3,4,5,6,7,8,9\}$.
Let R be the relation on A defined by $\left\{\right(x,y):x\in A,y\in Aandxdividesy\}$
Find (i) R (ii) domain of R (iii) range of R (iv) $R^{-1}$
State whether or not R is (a) reflexive (b) symmetric (c) transitive

Answer 1

Here, x R y if x divides y, therefore,
(i) $R=\left\{\right(2,2),(2,4),(2,6),(2,8),(3,3),(3,6),(3,9),(4,4),(4,8),(5,5),(6,6),(7,7),(8,8),(9,9\left)\right\}$
(ii) Domain of R $=\{2,3,4,5,6,7,8,9\}=A$
(iii) Range of R $=\{2,3,4,5,6,7,8,9\}=A$

(iv) $R^{-1}=(y,x):(x,y)inR=(2,2),(4,2),(6,2),(8,2),(3,3),(6,3),(9,3),(4,4),(4,8),(5,5),(6,6),(7,7),(8,8),(9,9)$

Infact $R^{-1}$ is $\left\{\right(y,x):x,y\in A,yisdivisiblebyx\}$
(a) As $(2,2),(3,3),(4,4),(5,5),(6,6),(7,7),(8,8)and(9,9)$ belong to R, therefore, R reflexive.
(b) Here, R is not symmetric. We may observe the $(2,4)\in R$ but $(4,2)\notin R$. Infact, 'x' divides 'y' does not imply 'y divides x' when $x\ne y.$
(c) As 'x' divides y' and 'y divides z' imply 'x divides z'. the therefore, the relation R is transitive.