Discrete Mathematics 2
Exam File
Spring 2011
Exam #1
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
Prove that
is irrational.
Let S be a set of six distinct positive integers whose maximum is at most 14. Show
that the sums of the elements in all the nonempty subsets of S cannot all be
distinct.
Suppose A = {a, b, c, d, e, f} and B = {x, y, z}.
a.)
Prove that f: A  B cannot be one-to-one.
b.)
How many functions from A to B are possible?
c.)
How many onto functions from A to B are possible?
Suppose C is the set of continuous functions from the [0, 1] to [0, 1]. Suppose f, g
 C. We say that f  g if and only if
< 0. Answer the following
questions. Be sure to explain your answer.
a.)
Is  reflexive?
b.)
Is  irreflexive?
c.)
Is  symmetric?
d.)
Is  transitive?
e.)
Is  antisymmetric?
f.)
Does  produce a partial order on [0, 1]?
g.)
Does  produce an equivalence relation on C?
Suppose f: A  B and g:B  C are one-to-one and onto functions.
a.)
Prove that g  f is one-to-one.
b.)
Prove that g  f is onto.
In your drawer you have 11 identical plaid socks, six identical blue socks, four
identical green socks and nine identical red socks. You grab socks out of the
drawer without looking.
a.)
How many socks must be pulled from the drawer to be certain of having a
matching pair of socks?
b.)
How many socks must be pulled from the drawer to be certain of having
four matching socks (my dog will be wearing them)?
c.)
How many socks must be pulled from the drawer to be certain of having a
matching pair of green or plaid socks?
Suppose A is the set of positive integers. For a, b  A, we define the relation, R,
by a R b if and only if a2 > b.
a.)
Determine if R is reflexive.
b.)
Determine if R is symmetric.
c.)
Determine if R is transitive.
d.)
Determine if R is antisymmetric.
e.)
Determine if R is an equivalence relation.
f.)
If R is an equivalence relation, find the equivalence classes.
Suppose R is an equivalence relation on a set A. For a, b  A, prove that [a] =
[b] or [a]  [b] = .
Suppose A = {a, b, c, d, e, f, g}.
10.)
11.)
12.)
13.)
14.)
15.)
a.)
How many relations are there on A?
b.)
How many relations on A are symmetric?
c.)
How many relations on A are reflexive?
d.)
How many relations on A are equivalence relations?
Give an example of a relation on the integers that is:
a.)
symmetric but not reflexive and not transitive
b.)
symmetric and transitive but not reflexive
c.)
reflexive and transitive but not symmetric
d.)
reflexive but not symmetric and not transitive
e.)
reflexive and symmetric but not transitive
f.)
reflexive, symmetric and transitive
g.)
transitive but neither reflexive nor symmetric
h.)
neither reflexive, symmetric nor transitive
How many equivalence relations are there on A = {a, b, c, d, e, f ,g ,h, i} that
produce 4 equivalence classes?
Seven couples are sitting at a circular table. How many seating arrangements
are there if:
a.)
Each couple sits together.
b.)
At least one couple sits together.
c.)
Exactly one couple sits together.
d.)
No couple sits together.
The 26 letters of the alphabet are put in a random order. How many orders are
there in which:
a.)
the word "facetiously" appears?
b.)
the word "facetiously" does NOT appear?
c.)
none of the words "dog," "cat," "fern" and "bus" appear.
Find (12360). [12360 = 23 ·3·5·103]
How many positive integers less than or equal to 12360 are NOT divisible by 2,
3, 5 or 7?
Exam #2
1.)
2.)
Suppose A is the set of integers. For a, b  A, we define the relation, R, by a R b
if and only if a2 = b2.
a.)
Determine if R is reflexive.
b.)
Determine if R is symmetric.
c.)
Determine if R is transitive.
d.)
Determine if R is antisymmetric.
e.)
Determine if R is an equivalence relation.
f.)
If R is an equivalence relation, find the equivalence classes.
Suppose A = {a, b, c, d, e, f}.
a.)
How many relations are there on A?
b.)
How many relations on A are symmetric?
c.)
How many relations on A are reflexive?
d.)
How many relations on A are equivalence relations?
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
11.)
12.)
13.)
14.)
15.)
16.)
Give an example of a relation on the integers that is:
a.)
symmetric but not reflexive and not transitive
b.)
reflexive and transitive but not symmetric
c.)
transitive but neither reflexive nor symmetric
How many equivalence relations are there on A = {a, b, c, d, e, f} that produce an
even number of equivalence classes?
You are taking a matching exam. There are five questions and five responses.
How many ways are there to answer the questions if there are:
a.)
no correct answers?
b.)
at least one correct answer?
c.)
exactly one correct answer?
Find (12600). [12600 = 23 ·32·52·7]
How many positive integers less than or equal to 12600 are NOT divisible by 2,
3, 11 or 13?
Find a recurrence relation, with initial condition, that uniquely determines the
following geometric progression.
2, 10, 50, 250, . . .
Find the unique solution for the following recurrence relation.
an+1 - 1.5 an = 0, n > 0
Solve the following recurrence relation.
2 an+2 - 11 an+1 + 5 an = 0, n > 0, a0 = 2, a1 = -8
Solve the following recurrence relation.
an+1 - an = 3n2 - n, n > 0, ao = 3
Draw two trees with 6 vertices that are not isomorphic.
For each sequence below, draw a graph with that as a degree sequence, if
possible. If not possible, explain why not.
a.)
1, 1, 1, 3
b.)
0, 0, 1, 1, 2
c.)
2, 3, 4, 5, 6
d.)
1, 1, 1, 1, 2, 3, 4
Draw a graph with 6 vertices that:
a.)
has an Euler circuit.
b.)
does not have an Euler circuit.
c.)
has an Euler trail.
d.)
does not have an Euler trail or circuit.
e.)
that is not connected.
f.)
that is 4-regular.
Construct the graph K4.
Can a tree have the following degree sequences? If not explain why not.
a.)
1, 1, 1, 3
b.)
0, 0, 1, 1, 2, 4
c.)
1, 1, 1, 1
d.)
3, 3, 3, 3
Exam #3
1.)
Solve the recurrence relation an + an-1 - 6an-2 = 0, n > 2, a0 = -1, a1 = 8.
2.)
Solve the recurrence relation an+2 = 4an+1 - 4an, n > 0, a0 = 1, a1 = 3.
3.)
Solve the recurrence relation an - an-1 = 3n2, n > 1, a0 = 7.
For #4-6, consider the following graphs.
4.)
5.)
6.)
7.)
8.)
9.)
Write the degree sequences for each.
a.)
graph 1
b.)
graph 2
Explain why graph 1 and graph 2 are not isomorphic.
For each graph, explain whether or not the graph has an Eulerian trail or an
Eulerian circuit. Give reasons for your answers.
a.)
graph 1
b.)
graph 2
For each sequence below, draw a graph (not multigraph, no loops) with that as a
degree sequence, if possible. If not possible, explain why not.
a.)
1, 1, 1, 1, 2, 3, 3
b.)
0, 0, 1, 1, 1
c.)
1, 1, 2, 3, 4, 5, 6
d.)
1, 1, 1, 1, 2, 3, 4
Draw a connected graph (no loops) with 6 vertices that: (do not use the same
graph more than once)
a.)
has an Euler circuit.
b.)
does not have an Euler circuit.
c.)
has an Euler trail.
d.)
does not have an Euler trail or circuit.
e.)
that is 4-regular.
Can a tree have the following degree sequences? If so, draw one. If not explain
why not.
a.)
1, 2, 2, 3
b.)
0, 0, 1, 1, 2, 4
c.)
1, 1, 2, 2
d.)
2, 2, 2, 2
Exam #4
1.)
2.)
3.)
4.)
5.)
6.)
7.)
graph 1
graph 2
graph 3
graph 4
For each of the graphs above, determine which are planar. Give reasons why or
why not.
For each of the graphs above, determine which are bipartite. Give reasons why
or why not.
For each of the graphs above, determine which are trees. Give reasons why or
why not.
For each of the graphs above, determine which admit a Hamilton Cycle. Give
reasons why or why not.
For each of the graphs above, determine the chromatic number.
For graphs 3 and 4, determine the Chromatic Polynomial.
Let U be a set. For any subsets, A and B of U, define "addition" (denoted by )
and "multiplication" (denoted by ʘ) as follows.
AB=AB
AʘB=AB
Determine if (U, , ʘ) is a ring.
Final Exam
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
Suppose R is an equivalence relation on a set A. For a, b  A, prove that [a] =
[b] or [a]  [b] = .
Prove that
is irrational.
How many positive integers less than or equal to 2500 are divisible by 5 or 11?
Suppose A = {a, b, c, d, e, f}.
a.)
How many relations are there on A?
b.)
How many relations on A are symmetric?
Find (12600). [12600 = 23 · 32 · 52 · 7]
Six couples are sitting at a circular table. How many seating arrangements are
there if:
a.)
Each couple sits together.
b.)
At least one couple sits together.
Can trees have the following degree sequences? If not explain why not. If so,
draw one.
a.)
1, 1, 1, 1, 4
b.)
0, 0, 1, 1, 2, 3
c.)
1, 1, 2, 2
d.)
1, 2, 2, 3
Suppose A = {a, b, c, d, e, f} and B = {x, y, z}.
a.)
Prove that f: A  B cannot be one-to-one.
b.)
How many functions from A to B are possible?
In your drawer you have 13 identical plaid socks, seven identical blue socks, two
identical green socks and four identical red socks. You grab socks out of the
drawer without looking.
a.)
How many socks must be pulled from the drawer to be certain of having a
matching pair of socks?
b.)
How many socks must be pulled from the drawer to be certain of having
four matching socks (my dog will be wearing them)?
c.)
How many socks must be pulled from the drawer to be certain of having a
matching pair of green or plaid socks?
For each of the following conditions, draw a graph, if possible, that satisfies the
conditions. If it is not possible, explain why.
a.)
planar and contains an Eulerian Path
b.)
a tree that is not planar
c.)
a bipartite graph with Chromatic Number 3