the following information is *tentative* for exam 2 and the final,
and will be updated closer to the exam date.

this exam will cover chapters 1-3 of hammack, excluding sections 1.8-1.10,
3.8 and 3.10. you should expect several true/false questions (no justification
and no partial credit), and several problems. for some of the problems you may
not *need* to show any work (but you will be graded on what you write,
so don't add nonsense to a correct answer, but do add some coherent thoughts
if you're not sure of your answer),
and for some you should explain your solutions clearly and logically in
complete sentences.

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

this exam will cover chapters 4-9 (excluding 8.4) and sec 10.1 of hammack.

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)
- be able to critique and correct bad proofs
- 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

**online rules and procedures:** you will download the exam from the
"Exam 2" assignment
in canvas after 10:25am on the day of the exam. you are allowed to
use the official course textbook (book of proof, 3rd ed, physical or digital)
during the exam, and email me if you have questions about the exam. no other
resources are allowed (no calculators etc, no notes or old homeworks, no
looking at the canvas discussions, no other websites etc, no contact with
others---if you have further questions please ask beforehand).
the exam will include an integrity pledge that you are to handwrite and sign
affirming that you are abiding by the exam rules. (there will be no
proctoring so we will rely upon an honor code.)
you are to then handwrite solutions. if you have a printer, you are welcome
to print out the exam and write directly on that. otherwise, use blank paper
and make sure your problems are clearly numbered and in order.
you should finish the exam by 11:20am (a couple minutes extra is fine) and
upload images of your exam (integrity pledge + solutions) to the "Exam 2"
assignment on canvas. this assignment will close at 11:30am, so i recommend
that you try to submit your exam several minutes early in case of technical
delays. (in the event of any technical issues, email me asap, and if possible
attach a copy of your exam.)

**online exam format:** due to the online nature of the exam, following
ou's recommended guidelines, i am going to try to design the exam to have
more "conceptual/critical" questions than i normally would. e.g., i might
ask about why a certain proof technique might be good or bad in some situation,
or i might ask you to analyze the quality/clarity of given proof. i will
also ask you more standard questions like we have been doing in class, but
as i don't have good source of such problems, and they are not so easy
for me to come up with, i don't currently have such practice problems for you
right now. hopefully i will come up with 1 or 2 on wednesday's discussion.
i will also try to make the exam shorter than it would be for an in-class exam,
both because i don't assume you have a distraction-free environment in which
to take the test, and i presume the process will be a little slower.
that said, even though the exam is open book, i recommend
you study well for the exam. if you need to read the book during the exam,
you will not have time to finish. you should aim to prepare so that
you don't need the book during the exam, but if you forget something then you
can use it to quickly look something up.

to help you review, here are some practice problems and my comment/hints from the last time i taught this course in 2018:

- 2018 exam 2 review problems (first try these on your own; also posted under the "exam 2 practice" assignment on canvas)
- my comments/hints to these review problems

i suggest you try these before discussion wed (apr 15), and then you can ask any question you may have.

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 14). 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), and be able to show if a set is countable or uncountable

to prepare, i suggest you begin by reviewing your midterm exams, and the recent homeworks and discussions. i also recommend you try many practice problems. you can try looking at examples/exercises from the book and the list of final review problems i made from the last time i taught the course:

- 2018 final review problems (first try these on your own; also posted under the "exam 2 practice" assignment on canvas)
- my comments/selected solutions to these review problems

**online rules and procedures:** the final exam will be administered
as a canvas quiz. it is a 2-hour exam which you
are to start at the time of your choosing between 8am and 2pm on the scheduled
exam day. that said there will be a 20-minute buffer to allow for technical
issues, which means you have 140 minutes from the time you start (though
no later than 4:20pm) until you
have to submit your answers. as with exam 2, there will be an honor code
(integrity pledge) in lieu of any proctoring. however, unlike exam 2,
there will be text boxes for your to type in your answers to each problem.
for some problems (think: problems where you need to show work or write a
proof), you will be allowed to either type everything in the text box
or handwrite your answer on paper. if you handwrite your answer, you can
either upload an image in the answer box for that question, or you
can attach all of your handwritten work as a **single file**
in the final attachments question.
please only submit pdf or image files, instead of something like a zip or tar file.
(leave the final attachments question blank
if you have no other attachments.)
there is a "mock final" quiz on canvas that you can try to see what this will
look like.

as for what is allowed, the rules are the same as for exam 2. you are allowed to use the official course textbook (book of proof, 3rd ed, physical or digital) during the exam, and email me if you have questions about the exam. no other resources are allowed (no calculators etc, no notes or old homeworks, no looking at the canvas discussions, no other websites etc, no contact with others---if you have further questions please ask beforehand).

**what to expect:** due to the online nature of the exam,
as with exam 2, i am going to try to design the exam include more
"conceptual/critical" questions than i normally would. e.g., things like
critique proofs, or maybe explain why you would use one proof technique
instead of another. as for the length, expect it to be about twice as
long as either midterm exam. you should prepare as you would for an in-class,
closed-book exam. in particular, do plenty of review problems to the point
where if you see a random exercise from the book, you know more-or-less
immediately how you should proceed, and that you only need to refer to
the book to double check some detail. (i don't expect you to quote
theorem numbers on the exam.)

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