Lecture 19
Exam: Tuesday June15 4-6pm
Overview
General Remarks
• Expect more questions than before that test your knowledge of the material.
(rather then deep insight). This means you should be able to get a good grade
if you study hard.
• Expect more questions on the material treated after the midterm, but there will
be questions on the material before midterm. This exam is to test your
knowledge and understanding of all the material treated in class.
• The examples in the book and homework assignments will serve as “inspiration”
for the questions in the exam.
• Do not just write down the answer to a question, also provide us with the
calculation and insights. For instance, if you are asked to write a recurrence
relation and then solve it, you can get full credit for the second part if you show
how you solved it even though the recurrence relation itself is wrong!
• Take a good look at the midterm and at the sample-exam that I treat next to get
an idea of the kind and level of questions you can expect.
Disclaimer
The following is a only study guide. You
need to know all the material treated in
class
1.1
•
•
•
•
Definitions: know all the terms involved.
Logical operators: how do they work?
Truth tables
Know how propositions are combined
using operators.
1.2
• Understand logical equivalence.
(what does it mean to prove one ?)
• De Morgan’s law
• See if you understand the simpler ones in
table 5.
1.3, 1.4
• Understand universal and existential
quantification and how to work with them.
• For instance: why is P(x) not a proposition
without a quantifier?
• Rules for negating quantified statements.
• see also midterm questions.
• Understand how nested quantifiers work
xyP ( x, y )
1.5
• Know the most important rules of inference by
heart: addition, simplification,
conjunction, modus ponens, modus tollens,
hypothetical syllogism.
• Know how prove a logical statement or
detect fallacies.
• Know the 3 most important methods of proof:
direct, indirect, by contradiction.
• You may be asked to prove simple propositions.
• What kind of theorems with quantifiers are there?
1.6
• Know all the definitions (e.g. empty set ,
power set, subset, cardinality, Cartesian
product etc.).
• Venn diagrams
1.7
• Know all the operations on sets (e.g.
intersection, union, disjoint, difference,
complement.
• Know some simple set identities treated in
text, like negation of a union is intersection
of negations.
1.8
• Understand what one-to-one, onto and
one-to-one associations are.
• Inversion, addition and multiplication and
composition of functions.
3.1 3.2
• Read 3.1 to train yourself in proving theorems.
You may be asked to prove or disprove a simple
theorem.
• Train yourself with sequences and summations.
Most important ones: geometric and arithmetic
progression
• Know what the solution is to a geom. and artihm.
summations. You may be asked to find the
solution of a summation using these.
• Definition of countable/uncountable: what does it
mean, can you prove a simple example.
3.3
• You can be asked to prove a simple
theorem by induction (see quiz): train
yourself.
• Difference induction-strong induction?
3.4
• What does it mean to define something
recursively (i.e. basis step, inductive step).
• How can we recursively define sets, such
as rooted, binary trees?
• Some material is excluded from this
section (see webpage).
4.1, 4.2
• Counting is difficult: it requires training!
(study all examples in book and homework
assignments)
• Product rule, Sum rule: know how to work
with them.
• Pigeonhole principle: understand what it
means.
4.3
• Permutations and Combinations (without
repetition, replacement).
• Look at slides: placing balls in baskets.
• You have to be able to recognize that a
particular problem is one of these cases:
e.g. find out if the “baskets” are distinguishable
or indistinguishable.
4.4
• Binomial theorem.
• Binomial coefficients
• You don’t have to learn the corollaries by
heart, but you need to have some practice
in manipulating binomial coefficients.
4.5
• Look again at slides: now there are 4 cases and
you have to be able to recognize a problem as
one of these 4 (balls and/or baskets can be
distinguishable/indistinguishable.
• Look at the examples, home-works, midterm,
sample final, quizzes. Practice!
• Theorem 3.
5.1,5.2
• Basic definitions: , event, sample space,
prob. of complement, prob. of union, prob.
of intersection.
• Non-uniform probabilities.
• conditional prob. independence. (e.g. you
may be asked if 2 events are independent).
• Bernoulli trials, Binomial distribution
(recognize that a problem is a Bernoulli trial)
• Random variables.
5.3
• Expected values and Variance, standard deviation
(you may be asked to compute them).
• Linearity of expectation. This trick may help you when
you are asked to compute expectation of sums of
random variables.
• Geometric Distribution: what does it model?
• Independence and implications for mean/variance
(they may simplify your calculations).
• Chebychev’s inequality.
6.1,6.2
• Recurrence Relations: How do you construct
one from a description (e.g. see quiz question
on bank interest).
• How do you solve one! (you may be asked to
solve “simple” recurrence relations of various
sorts: e.g. with the same roots, with or without
initial conditions etc., see sample exam).
• If you study the material in the book and practice
there should be no surprises for you here.
6.4
• What is a generating function. You should be able to
construct one given a sequence and vice versa.
• Combining generating functions (add & multiply).
• Extended binomial coefficients (definition).
• Learn by heart GenFunc for 1/(1-ax), (1+x)^u (th.2).
• Study examples on how they are used to solve
counting problems with constraints and recurrence
relations.
6.5, 6.6
• Understand and know by heart the formula
for inclusion/exclusion.
• Understand how it is applied to counting
problems of the sort: count the number of
elements that do not have a the following
properties.
• Derangements: what is it and how many
are there?