Question

Let R be a relation from A = {1, 2, 3, 4} to B = {1, 3, 5} such that
R = [(a, b) : a < b, where a $\epsilon$ A and b $\epsilon$ B].
What is RoR${}^{-1}$ equal to?

• A. $(1,3),(1,5),(2,3),(2,5),(3,5),(4,5)$
• B. $(3,1),(5,1),(3,2),(5,2),(5,3),(5,4)$
• C. $(3,3),(3,5),(5,3),(5,5)$
• D. $(3,3),(3,4),(4,5)$

Answer 1

Answer: (C)

$(3,3),(3,5),(5,3),(5,5)$

$R$ gives the set $\left\{\right(1,3),(1,5),(2,3),(2,5),(3,5),(4,5\left)\right\}$

$R^{-1}$ gives the set $\left\{\right(3,1),(5,1),(3,2),(5,2),(5,3),(5,4),(3,3\left)\right\}$

To compute $Ro{R}^{-1}$, we pick 1 element from $R^{-1}$ and its corresponding relation from $R$.

eg: $(3,1)\in R^{-1}$ and $(1,3)\in R⟹(3,3)\in Ro{R}^{-1}$

Similarly, computing for all such pairs, we have

$Ro{R}^{-1}=\left\{\right(3,3),(3,5),(5,3),(5,5\left)\right\}$