Test 2
Finite Mathematics
MAT115
Fall 2007
Time duration: One hour, Test date: Oct 23, 2007
Test modality: Closed book, no note, no laptop. Cell phone must be turned off during the exam
time.
Answer only 5 questions from below
1. Let R be a subset of the product set A  A . Under what condition(s), could one claim
R to be (a) reflexive, (b) symmetric and (c) transitive, respectively? Assume
x, y  Horses and xRy means x runs faster than y. Indicate, with reason, which of the
following from each pair comprise a valid description of R :
a. Reflexive or irreflexive
b. Symmetric or anti-symmetric
c. Transitive or intransitive
2. Consider a relation R on the set A  A . Let R  {(1,1), (1,3), (3,1), (2,2), (3,2), (3,3)} .
Describe the relation as a matrix M (or in a tabular form). How does the matrix form of
the inverse relation R 1 appear? Consider a relation T on A  A . Assume, T is a
symmetric relation. Show that T 1 must be symmetric as well.
3. Consider a universal set comprising people. For each of the following scenarios, decide
whether or not the underlying relation in each case is transitive.
a.
b.
c.
d.
xRy
xRy
xRy
xRy
means
means
means
means
x and y have same parents
x and y have different parents.
x is taller than y
x is not taller than y
Consider the situation where the relation xRy means x is father of y . Is it transitive,
intransitive, or neither? If ~ R is the relation “not father of”, how do you categorize
x ~ Ry ? Is it transitive, intransitive, or neither?
4. Consider the relations R and T on cities such that xRy the cities x and y are both on
the same side (east or west) of Greenwich Meridian, and xTy means x and y are both
on the same side of the equator (north or south). Show that these two relations are
equivalence relations. Using these two relations, obtain the partition of the set of cities C
where C = {Bangkok, Cairo, Oslo, Rio de Janeiro, Utica, Mexico City, La Paz, Sydney,
Tokyo, Dublin}
5. Consider two sets A and B . Under what condition can the function f : A  B be
legitimate? Assume both f and f
1
exist. If f : A  B then on what does f
1
operate?
1
What would be the criterion (or criteria) for f to exist? Is there any difference between
a function and a relation? Are all relations functions? Are all functions relations? Explain.
6. Consider the following function. Let Z  be a set of all positive integers and
f : Z   Z  with f (i)  i div 2  1 where div is an integer division operator. In
other words, if you compute i div k you’d get either 0 as a result or it would be the
largest integer q such that kq  l  i where l is the remainder after division. What kind
of function is f ? Consider a function g on a character domain C ={ a, b, c, d, .., x, y, z}
such that
 Z if l  a 
g (l )  

 A, if l  a 
What kind of function is g ?
7. Sketch the following function and determine its domain and range:
y  x3  4x 2  x  6
8.
Sketch the following function and determine its domain and range:
y
( x  3)
( x  2)( x  6)