Consider the following relation R on A where A = {1, 2, 3, 4, 5} aRb ⇔ a b < min(a, b) For example, 2R4 since 2 4 = 1 2 and min(2, 4) = 2 and 1 2 < 2. (a) Draw the digraph of R (4) (b) Give a path of length 2 from 3, if any (2) (c) Give the domain and range of R. (4) (d) Determine R(2)

Solution:

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

aRb ⇔(a,b) if $\frac ab<min(a,b)$

(a) R = {(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}

Digraph of R:

(b) We need a path of length of 2 from 3. It means a = 3. When a=3, b is either 4 or 5 only from set R, i.e., (3,4), (3,5).

From (3,4), we find only one point of length 2 from above graph, which is (1,4).

From (3,5), we find only one point of length 2 from above graph, which is (1,5).

(c) For any set X = {(a,b)}, Domain is {a}, Range is {b}.

So, domain of R = {1,2,3,4}, Range of R = {2,3,4,5} from part (a).

(d) R = {(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}