discrete math - exam info
exam 1 (wed oct 3, in class)
this exam will cover chapters 1-3 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
in class friday and monday, i will give you practice problems
(with an emphasis on the material from chapters 1 and 2, since counting
should be fresher in your minds) to help prepare. i suggest you study
as much as possible before these classes so you can treat these practice
problems as mock exams and see where you need to improve.
exam 2 (mon nov 19, in class)
this exam will cover chapters 4-9 (excluding 8.4) and sec "10.0" (pp 154-160)
of hammack. (e.g., no strong induction or fibonacci numbers, though you are
allowed to use strong induction or smallest counterexample method on the
exam if you wish. and this material may appear on the final exam.)
here 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
to help you prepare, here are some practice problems that i recommend you
study for and try on your own (like a mock exam, but longer), and
bring any questions you have to class friday before the exam. there is also
a version with comments/hints for you to help check your solutions.
(if you spot any typos, let me know so i can correct them)
final exam: wed dec 12 (8-10am)
the final exam will be cumulative, covering everything on exams 1 and 2,
as well as some aspects of functions and cardinality (ch 12, ch 13).
specifically, you should be comfortable with the following:
- all topics listed above for exams 1 and 2
- be able to construct functions from one set to another, state the
domain, codomain and range (image)
- prove that a function is injective/surjective/bijective or that it is not
- count the number of functions between 2 finite sets possibly
with certain properties (e.g., those that are injective)
- show that 2 infinite sets have the same cardinality
- know examples of infinite sets with different cardinalities (like Z and R)
some suggestions for problems to review for the final are:
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