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 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

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.

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?