Transcript Lecture19Overview05

Lecture 19
Exam: Tuesday June14 4-6pm
Overview
Disclaimer
The following is a only study guide. You
need to know all the material treated in
class
1.1
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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.
• 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 correspondence 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 slides).
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).
• mean and variance of Binomial distribution.
6.1,6.2
• Recurrence Relations: How do you construct
one from a description.
• 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).
• 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?