exam 1 (mon 3/6, in class)

this exam will be on the topics we covered from the beginning of the semester through the end of february. this consists of most of chapters 1, 2 and 3 of hammack, and most of parts I and II of houston. (notable exclusions are the material on functions from chapter 1 of houston and the last 3 sections of chapter 1 of hammack.)) here are some things which i consider important, but this is not necessarily an exclusive list (not necessarily a sufficient list?) of topics for the exam:

• be able to explain your solutions in coherent english/math and write in complete sentences.
• understand basic set constructions (union, intersection, difference, power set, complement) as well as notation
• be able to determine if an object is an element or a subset of another set (perhaps of a set constructed by methods mentioned above)
• be able to determine the cardinality of a finite set (perhaps of a set constructed by methods mentioned above)
• understand basic statement constructions (and, or, negation, conditionals, inverse, converse, contrapositive) as well as notation
• understand the use of quantifiers
• determine if two statements are logically equivalent, or if one implies the other
• basic counting, binomial theorem, and the inclusion-exclusion principle

to help you prepare, or just for your records, here's a pdf of worksheets 1-7. the first 6 are ones we worked on (though didn't necessarily finish) in class, and the last worksheet, in 2-parts, has a bunch of practice problems for you to try. i suggest you work on these this week, and bring any questions you may have to class friday.

exam 2 (fri apr 21, in class)

this exam will cover all proof techniques we've studied since exam 1. this correponds to chapters 4-10 of hammack, excluding a few sections that we skipped (7.4, 8.4, 10.1, 10.2). the classes prior to exam 2 we will work on problems to review and solidify your ability to do an write proofs, including doing readings from houston. here are some more specific things you should be comfortable with:

• be able to write proofs clearly and coherently
• be comfortable with the following basic proof techniques: direct, cases, contrapositive, contradiction, and induction
• be able to disprove statements (counterexample)
• know how to do proofs about equality and containment of sets
• know how to do proofs about implications (conditionals) and logical equivalences (if-and-only-if proofs)
• given a true statement, be able to determine what proof techniques are suitable for proving it
• given a statement be able to determine whether it is true or false and prove or disprove it

besides the reviewing we will do in class, some other suggestions to prepare for the exams are: re/read the relevant sections in houston (part iv excluding chapter 25) and/or hammack, do practice problems on your own, and have someone look over your answers.

final exam: mon may 8 (8-10am)

the final exam will be cumulative. specifically, you should plan to be tested on all topics covered in the first 2 exams, as well as the material from chapters 11 and 12 of hammack (relations and functions). i expect that most of the exam will be on the material covered in the first 2 exams (probably at least two-thirds), with the topics from exam 2 (proofs) making up the largest portion of the exam. in terms of format, you should expect a number of true/false problems (no justification needed), a few free-form problems (e.g., counting problems, or questions about relations/funtions), and a few proof problems (which may be of the form "prove or disprove").

you should be comfortable with the following:

• all bullet points listed above for exams 1 and 2
• be able to determine if something is a relation and why. similarly for equivalence relation and function
• be able to determine (with proof) what properties (reflexive, symmetric, transitive) a relation has or does not have and why
• understand the relationship between equivalence relations and partitions
• do arithmetic with integers mod n
• given a function, be able to determine the domain, codomain, and range, as well as images and preimages of elements or sets
• be able to determine (with proof) if a function if injective/surjective/bijective
• be able to determine the inverse of a bijective function
• count relations/functions satisfying various properties
• know the definitions of all the above terms/properties for relations, functions

since we didn't have a proper problem set on relations and functions, here are some selected practice exercises from hammack chapters 11 and 12:
11.0: 7, 11
11.1: 12, 15
11.2: 9, 13
11.4: 7bd, 8
12.1: 5, 10, 11
12.2: 7, 15, 17
12.3: 5
12.4: 9
12.5: 9
12.6: 4, 7, 11, 12