Lecture 19 Exam: Tuesday June15 4-6pm

Lecture 19Exam: 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

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?

Lecture 19 Exam: Tuesday June15 4-6pm