Sociology 375
Exam 1
Spring 2005
Prof Montgomery
Answer all questions.
200 points possible.
1) [40 points]
Consider a relation R on a set S.
Slightly modifying the definition given
in lecture, suppose that R is
complete
iff
¬
(xRy
∨
yRx)
→
(x=y)
for all (x,y)
∈
S
×
S
.
a) For each of the following conditions, state whether the condition is equivalent to the
definition of completeness given above, and then explain why (or why not) using truth
tables.
[HINT: To save time, you might construct one large truth table to answer all 3
parts simultaneously.]
i)
xRy
∨
yRx
∨
(x=y)
for all (x,y)
∈
S
×
S
ii)
(xRy
∨
yRx)
→
¬
(x=y)
for all (x,y)
∈
S
×
S
iii)
xRy
∨
yRx
for all distinct (x,y)
∈
S
×
S
b) Consider the relation R = {(1,1), (1,2), (2,2), (2,3), (3,2)} on S = {1, 2, 3}.
Using the
definition of completeness given above, determine whether R is complete by testing all
ordered pairs.
[NOTE: To receive full credit, you must demonstrate that you have tested
all
ordered pairs.
You can show this by constructing a table in which the rows
correspond to ordered pairs, the columns correspond to propositions in the definition of
completeness, and the entries in the table are truth values.]
2) [60 points]
Consider relation R = {(1,4), (2,1), (2,3), (2,4), (4,3)} on S = {1, 2, 3, 4}.
a) Show how the relation R can be represented as
(i) an adjacency matrix A
(ii) a (directed) graph
b) In general, given a relation represented as directed graph, how can you tell (using the
graph) whether the relation is asymmetric?
In the present example, is the relation R
asymmetric?
If not, list any violations of the asymmetry condition.
c) State the matrix test for transitivity, and then show whether A passes this test.
If not,
list any violations of the transitivity condition.
d) Is R a strict partial order?
If it is, then draw the Hasse diagram.
If not, find the
transitive closure of R, and then draw the Hasse diagram of the transitive closure.
e) Using the adjacency matrix A, compute the Bonacich centrality vector given
α
= 1 and
arbitrary
β
> 0.
[HINT: There are no 4-paths in A.]