Sociology 375
Exam 1
Spring 2007
Prof Montgomery
Answer all questions.
250 points possible.
Explanations can be brief.
1) [40 points]
Given a binary vector v (with elements equal to 0 or 1, which may be
interpreted as truth values), we could use matlab to compute the truth values of the
following eight expressions:
all(v), all(~v), ~all(v), ~all(~v), any(v), any(~v), ~any(v), ~any(~v).
[HINT: Recall that ~ is the logical “not” operator in matlab.
Thus, the truth values in the
vector v are reversed in the vector ~v, and the truth value of all(v) is reversed by ~all(v).]
Assuming that the binary vector v has only 2 elements (each of which could be either
“true” or “false”), use a truth table to determine which of the preceding eight expressions
are logically equivalent (producing the same answer regardless of the truth values in v).
[HINT: Because v has only two elements, the vector v could be written as v = [P Q]
where P and Q are truth values, the all command becomes the logical “and” operator, and
the any command becomes the logical “or” operator.]
To receive full credit, you must
show the truth table and partition the eight expressions into equivalence classes.
2) [90 points]
Consider the relation A (‘sends information to’) on a set of four actors S =
{1,2,3,4}.
In particular, suppose that 1 sends information to 2, 2 sends information to 1,
2 sends information to 3, 3 sends information to 4, and 4 sends information to 3.
a) Show how the relation A could be represented as
i) a set
ii) a directed graph
iii) an adjacency matrix
b) Compute
i) the number of 3-paths between each ordered pair of actors
ii) the reachability matrix
iii) the distance matrix
c) In this example, is the reachability relation an equivalence relation?
If so, find the
equivalence classes generated by this relation.
If not, explain why.
d) Give the adjacency matrix for the relation ‘can be reached by.’
Forming the
intersection of the relations ‘can reach’ and ‘can be reached by’, we obtain the relation
‘can reach
and
be reached by’.
Is this relation an equivalence relation?
If so, find the
equivalence classes generated by this relation.
If not, explain why .
e) Suppose that we partition the set of actors into those actors who can be reached by all
others and those actors who cannot.
Which actors are in the first subset?
Which are in
the second subset?
Is this partition a regular equivalence? If not, briefly explain why.