Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m,...
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)?
Homework Answers
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