Lecture 19Exam: 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

• 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?