CS2022/ MA 2201 Homework #6
Due Monday, 12/13
#1. Let R = {(a,a), (b,c), (a,c)} be a relation on the set {a, b, c}. Show
a) Reflexive-closure(R)
{(a,a), (b,c), (a,c) (b,b), (c,c)}
b) Symmetric-closure(R)
{(a,a), (b,c), (a,c) (c,a), (c,b)}
c) Transitive-closure(R)
{(a,a), (b,c), (a,c)} (Already transitive)
d) The smallest equivalence relation containing R
{(a,a), (b,c), (a,c) (c,a), (c,b) (b,b), (c,c) ,(b,a), (a,b)}
#2. If R is the relation {(a,b), (a,c)}, S is the relation {(b,b)}, and T is the relation
{(b,c),(c,b)} on {a,b,c}, show:
a) R o S = 
But,
S o R = {(a,b)}
b) R o T = 
c) Show graphs of parts a and b
SoR: a  b  b
R o T: b c 
and ac 
#3. True or False:
a) If R is transitive, then Rn is transitive
Circle One: True
False
b) If R is transitive, then Reflexive-closure(R) is transitive.
Circle One: True
c) If R is reflexive then Symmetric-closure(R) is transitive
False
Circle One: True
False
#4. For the relation R = { (a,b), (b,c) }and for r = reflexive-closure, s = symmetricclosure and t = transitive-closure show that
r s (R) = s r (R)
s(R) = {(a,b), (b,a), (b,c), (c,b)}
r(s(R) = {(a,b), (b,a), (b,c), (c,b), (a,a), (b,b), (c,c)}
s(r (R) = {(a,b), (b,c), (a,a), (b,b), (c,c), (b,a), (c,b)}
These are identical sets.
#5. Show that the following are or are not equivalence relations:
a) {(a,b): a + b is even} over the positive integers, Z+
Reflexive
a + a is always even for all integers a
Symmetric
If a + b is even, then b + a is even
Transitive
We need to show that if a + b is even and b + c is even, then a + c is even.
If a + b even  Case 1) a odd and b odd or Case 2) a even and b even
If b + c even
Case 1: a, b odd; so c must be odd; therefore a + c is even
Case 2: a, b even; so c must be even; therefore a + c is even
So {(a,b): a + b is even} is transitive
It is Reflexive, Symmetric and Transitive:
So it is an Equivalence Relation
b) {(a,b): a + b odd} over the positive integers, Z+
This isn’t even reflexive, so it cannot be an equivalence relation
#6. Solve the vegetarian and cannibal problem (graphically) for the case where there are
3 vegetarians and 3 cannibals to be transported from one side of the river to the other.
In the following, we just show a successful set of moves. Bur remember, you need to
try all combinations of unique moves to solve this graphically.
#7. Draw the following graphs or state why they cannot be drawn:
a) a simple graph with 4 vertices of orders 1, 2, 3 and 4
Not possible. If there were such a graph, then the vertex of degree 4 would have to
be connected to 4 other nodes since in a simple graph there are no self-loops.
Also, in a simple graph, we had a theorem that stated that any simple graph of
degree > 2 has 2 vertices of the same degree.
b) a graph with 4 vertices of orders 1, 2, 3 and 4
#8. In a group of 15 people is it possible for each person to be acquainted with exactly 3
others? Why or why not?
Suppose in a group of 15 people each had exactly 3 acquaintances. Then if vertices
represented people, the graph would have 15 vertices of odd degree, and we know
the number of vertices of odd degree is even. Also, the total degree would be 45,
contradicting the fact that the total degree is even.
#9. Show a bipartite graph for the following graph:
The two disjoint sets are: S1={v1,v3,v5} and S2={v2,v4,v6}. No edges join vertices in
set S1 to vertices in the same set as well as no edges join vertices in S2 to vertices in
the same set.
#10. a) Create the graph which models the following map. b) Show the minimum number
of colors needed to color it:
Can be colored in 3 colors. Lots of ways to do it.