Question

Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3}

a. List all numbers of S.

b. List all ordered pairs in R.

c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)?

d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges.

e. Give each edge in the digraph G a name, then construct a table for the function γ : E(G) → V (G) × V (G).

f. Give a closed path of length 1, 2, 3, and 4. (vertices may repeat here)

g. Give a cycle of length ≥ 1, if it exists in G. (vertices may not repeat here)

h. Is the digraph G acyclic? (Example 3, pg 102)

i. How many loops (closed paths of the form vivi) does digraph G have?

j. Give the converse relation R←.

k. Give the matrix representation of R←.

l. Does R← satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)?

Answer 1

est S={nen/ een co} and R= {(min) e sxs I mem mod 3} @ S= {1,2,3,4,5) ), (2,2), (3,3), (4,4), (5,5), (1,4),(4,1), (2,5), (5,2)} - O R is reflexive because all (1,1), (2, 2), (3, 3), 14,41 (5,5) ER R is symmetric because (14) ER = (4,1) ER (215) ER =) (5,2) ER since R is Reflexive, it is not anti-reflexive (AR). R is not anti- symmetric because (0,4), 14,1) ER but ! # 4. satisfies that (1,4), (4,1) ER it is transitive as -> (VER - and (2,5), (5,2) ER =) (2,2) ER . rence a satisfies (R), (5), (T) mot satisfies (AR), (AS) Scanned with CamScanner

O G G G GG Scanned with CamScanner