Sociology 375
Exam 1
Fall 2007
Prof Montgomery
Answer all questions.
250 points possible.
1) [30 points]
The antisymmetry condition can be written as
(xRy
∧
yRx)
→
(x = y)
.
Using a truth table, determine whether the condition
¬
(x = y)
∨
¬
(xRy)
∨
¬
(yRx)
is equivalent to the antisymmetry condition.
[HINT: Applying some elementary rules of
logic, you could potentially answer this question without a truth table, but you must
construct the truth table to receive credit on this question.]
2) [20 points] a) State the (set-theoretic) conditions for symmetry and asymmetry.
b) Consider the empty relation R =
∅
on a non-empty set S.
Describe (in words) how
this relation would appear if represented as a graph or as an adjacency matrix.
Is this
relation symmetric, asymmetric, both, or neither?
Explain why.
[HINT: Use the
conditions that you stated in part (a).]
3) [90 points] Consider the relation R = {(1,2), (1,3), (3,2), (3,4), (4,2)} on the set
S = {1, 2, 3, 4}.
a) Show how the relation R could be represented
i) as a (directed) graph
ii) as an adjacency matrix
iii) using infix notation
b) Determine (by computation or inspection)
i) the number of 2-paths between each pair of individuals
ii) the number of 3-paths between each pair of individuals
iii) the reachability matrix
iv) the distance matrix
c) State the matrix test for transitivity.
Using this test, determine whether the relation R
is transitive.
If not, list any violations of transitivity.
d) Give the transitive closure of R.
Is the transitive closure of R an equivalence relation?
If not, explain why.
If so, give the equivalence classes.
Is the transitive closure of R a
strict partial order?
If not, explain why.
If so, draw the Hasse diagram.
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