Math 10 Discrete Mathematics
T. Henson
FINAL EXAM STUDY GUIDE
The final exam is comprehensive. Failure to take the final exam will result in a grade of ``F". You may
have a 5” X 8” help sheet (equivalent to a half-sheet of paper) for the final. Only one side of the paper
may be used. The ``help sheet" must be hand-written and must be turned in with your exam. The final
exam will be given on Friday, May 23, 5:15 – 7:45 pm.
Approximately 25% of the final exam will cover chapter 9.
Here is a summary of the material you should know for the final.
CHAPTERS TWO and THREE: The Logic of compound Statements and Quantified Statements;
Digital Logic Circuits
 simplify and negate propositions and quantified predicates;
 translate a written sentence into symbolic form, negate it and give a natural English translation.
 given a digital logic circuit,
o construct the input/output table of the circuit
o construct the Boolean expression that represents the circuit
o simplify the Boolean expression
o construct the simplified circuit
CHAPTERS FOUR and FIVE: Elementary Number Theory, Methods of Proof, Sequences and
Mathematical Induction
 use both direct and indirect proof techniques, including proof by cases, proof by contraposition
and proof by contradiction;
 write formal proofs using complete English sentences and principles of logic, rules of inference
and quantification;
 prove statements about integer, rational or irrational numbers;
 prove statements about divisibility.
 use the principle of mathematical induction to prove statements about sequences, divisibility
properties or inequalities
 use mathematical induction to prove a formula correctly represents a sequence defined by a
recurrence relation
 given a sequence, find a recurrence relation that represents the sequence
CHAPTER SIX: Set Theory
 apply principles and definitions of set theory to find unions, intersections, and set complement
and to determine whether one set is a subset of another;
 prove statements about sets
o Prove statements involving the subset relationship or set equality
o Use a contradiction argument to prove a statement involving the empty set
o Give an algebra of sets proof of a set identity
CHAPTER SEVEN: Functions
 prove or disprove a function is one-to-one or onto;
 find the composition of functions.
Final Exam Study Guide and Chapter Nine Review Exercises
Page 2 of 4
CHAPTER EIGHT: Relations
 Determine whether a relation is reflexive, symmetric, antisymmetric or transitive
 Determine whether a relation is an equivalence relation or a partial order relation
o If an equivalence relation, find the equivalence classes of the relation using correct
equivalence class notation
o If a partial order relation, sketch the Hasse diagram of the relation
CHAPTER NINE: Counting and Probability
 solve counting problems using elementary counting techniques
o sum and product rules;
o combination and permutation;
o principle of inclusion/exclusion
 Find the probability of an event
CHAPTER TEN: Graphs and Trees
 demonstrate a knowledge of basic graph theory terminology;
 apply graph theory to find sketch a graph
 demonstrate a knowledge of basic tree theory terminology;
REVIEW EXERCISES FOR CHAPTER NINE
While these problems are representative of the types of problems that I might put on an exam, they are
not inclusive. You should be prepared to work any type of problem related to the material studied.
NB: For all counting problems, indicate the counting principle(s) being used to solve the problem and
indicate your reasoning. On the exam NO CREDIT will be given for unsupported numbers.
1.
In a certain state every license plate has three letters followed by three numbers. There are 20
million cars in the state. Are there enough license plates to go around?
2.
Iota automobiles come in 4 models, 12 colors, 3 engine sizes and 2 transmission types.
a. How many distinct Iota automobiles can be manufactured?
b. If one of the available colors is lavender, how many different lavender Iotas can be
manufactured?
c. If one engine size is V-8, how many distinct lavender Iotas have a V-8 engine?
3.
A visitor to Raskin-Bobbin’s Ice Cream Shoppe may order a 2-scoop sundae by selecting 1 flavor
of syrup, one or two flavors of ice cream, whip cream or not, nuts, or not, and a cherry on top, or
not. There are 29 flavors of ice cream and 7 flavors of syrup. How many different 2-scoop sundaes
may be ordered?
4.
Eight horses are entered in a race in which a first, second, and third prize will be awarded.
Assuming no ties, how many different outcomes are possible?
Math 10 Discrete Mathematics
T. Henson
Final Exam Study Guide and Chapter Nine Review Exercises
Page 3 of 4
5.
Given four bands, seven floats and three equestrian units,
a. how many possible parades are there?
b. Suppose they are going to parade down the street with the bands first, then the floats and then the
equestrian units. How many such parades are possible?
6.
A professor has seven different mathematics books on her shelf. Four of the books deal with
Calculus and three are concerned with Discrete Mathematics. In how many ways can the professor
arrange the books on the shelf if,
a. There are no restrictions?
b. All of the Calculus books must be next to each other?
c. All of the Calculus books must be next to each other and all of the Discrete Mathematics books
must be next to each other?
d. In how many ways can the professor arrange any four of the books?
7.
Determine the number of six-digit integers (no leading zeros) in which
a. No digit may be repeated
b. Digits may be repeated
c. No digit may be repeated and the integer is even (Hint: consider two cases: the number ends in
zero or it doesn’t)
d. Digits may be repeated and the integer is even
8.
What is the value of k after the following code has been executed?
k:=0
for i:=1 to 100
for j:= 1 to 300
[Statements in body of inner loop.
None contain branching statements
that lead outside the loop.]
k:=k+1
9.
A student council consists of 15 students.
a. In how many ways can a committee of six be selected from the membership of the council?
b. Suppose the council consists of eight women and seven men.
i.
How many committees of six contain exactly three women and three men?
ii. How many committees of six contain at least four women?
c. In how many ways can a president, vice-president, and treasurer of the council be selected from
the members of the council?
10. Three officers – a president, a treasurer, and a secretary – are to be chosen from among four people:
Lucy, Desi, Ethel and Fred. The following conditions must be satisfied in selecting the officers:
 Desi cannot be president (but he can hold any other position)
 Either Lucy or Ethel must be secretary (the one who is not secretary can hold another
position).
In how many ways can thee officers be chosen?
Math 10 Discrete Mathematics
T. Henson
Final Exam Study Guide and Chapter Nine Review Exercises
Page 4 of 4
11. A student is to answer seven out of ten questions on an examination. In how many ways can she
make her selection if
a. There are no restrictions?
b. She must answer the first two questions?
12. Find the number of 5-card poker hands containing
a. The ace of spades and the ace of diamonds
b. The ace of spades and the ace of diamonds but no other aces
13. Among the 40 first-time campers at Camp Forlorn one week, 14 fell into the lake during the week,
13 suffered from poison ivy, and 16 got lost while trying to find the dining hall. Three of these
campers had poison ivy rash and fell into the lake, 5 fell into the lake and got lost, 8 had poison ivy
and got lost, and 2 experienced all three misfortunes. How many first-time campers got through the
week without suffering any of these mishaps?
14. Among a group of 200 college students, 19 study French, 10 study German, and 28 study Spanish.
If 3 study both French and German, 8 study both French and Spanish, 4 study both German and
Spanish, and 1 studies all three languages, how many of these students are
a. NOT studying any of these three languages?
b. Studying Spanish only?
15. An experiment consists of rolling a pair of dice, one red and one green, and observing the outcome.
a. What is the sample space for this experiment? How many elements are in the sample space?
b. Find the following probabilities:
i. the probability that the same number turns up on each die (e.g. one pip on the red die and one
pip on the green die).
ii. The probability that the sum of the pips on the two dice is 7.
iii. The probability that the sum of the pips on the two dice is at most 5
16. Among a box of 14 cell phones, 5 are broken.
a. If 4 cell phones are to be selected, how many selections are possible in which two cell phones are
broken?
b. If 7 cell phones are to be selected, how many such selections are possible in which at most 2 of
the cell phones are broken?
c. Suppose three cell phones are to be selected at random. What is an appropriate sample space for
this experiment? (Describe in words) How many elements are in the sample space?
d. What is the probability that of the 3 cell phones selected at random 2 are broken?
17. The board of directors of a corporation has 10 members, of whom 6 are women and 4 are men. A
delegation of 4 members is to be selected randomly.
a. What is an appropriate sample space for this experiment? (Describe in words) How many
elements are in the sample space?
b. What is the probability that the delegation will consist of women only?
c. What is the probability that the delegation will have exactly two men and two women?
d. What is the probability that the delegation will have at least two women?
Math 10 Discrete Mathematics
T. Henson