(Due on the desk at the front of class,
between 6:50 and 7:00 p.m., Tuesday, 12 Feb.)
Just two reasonably short proofs for now. I'll add a third
problem by 3 p.m. Thursday.
New rules for ALL assignments in MATH 1090:
Keep the old rules -- everything posted on the course page earlier
in the course. But also follow these rules:
1. Staple sheets together at the upper left corner if you submit
more than one page for an assignment. (No paper clips, no string, no
tape, etc. Staple.) No ratty tattered paper at one side because of
ripping pages out of a spiral notebook, either. Scissor the ratty
stuff off before handing in the work.
2. Underline or CAPITALIZE your last name. The grader should not
have to spend any time at all figuring out which is your last name
if you write
(a very popular name (not) in Sri Lanka)
as your name on a paper. And it makes a big difference, where your
paper ends up in an alphabetized stack of papers to be returned in
Jamessir Bensonmum is not good enough. Is it
Jamessir BENSONMUM or JAMESSIR, Bensonmum?
Of course, anyone familiar with Indian names would
know that Vaidyanathaswamy is a surname, as is Abhyankar,
and that Thiruchirappalli is a city in Tamil Nadu.
But hardly any North American graders can be expected
to know such things. Similar remarks apply to names
from many countries. Moreover, in many countries a
name can serve as either a first name or a surname.
In those cases NO ONE can tell which is the surname, in
the absence of information supplied by the name's
owner. So: Supply it.
3. If the paper is a submission from a group with two or more
members, CIRCLE the name of the person who will be responsible
for picking up the paper in class when it is graded and returned.
Also put an outline around the names of ALL group members, which
must be listed on the first page of the paper, if the group has at
least two members. E.g., draw a boundary around the entire clump
of three names
if those are the members of the group, and also circle individually
the second name if Ms. BARRY is to pick up the graded paper.
4. DOUBLE-SPACE ALL WORK on all assignment papers.
And leave at least 3 cm margins on each
side (left, right) of the page. These spaces are
meant to give the grader room for marks and comments.
jam all your work together on the page, single-spaced, no margins; this
makes it impossible for a grader to deal with your paper and also puts
her/him in a bad mood when grading your paper, which you don't want.
I will instruct the grader to deduct marks from the papers of
people who do not follow these rules. Not following reasonable
rules shows disrespect to the people who make them and the people
(in this case, the grader and the instructor) whom the rules are
meant to help.
See the comments below the problems before starting the problems.
Prove that |- A \/ B == A \/ ¬B == A .
Give a bullet (equational) proof that the above wff lives in the
room of theorems, no matter what the wffs A and B are. You may
use, in your proof, any axioms from (1) through (11), and any theorems
up through 2.4.10 listed in the list of theorem schemes posted on the
course page last week. In fact, always, if nothing is said to the
contrary, you may use all axiom schemes, (1) - (11), in proofs.
Give a bullet proof that |- A /\ (A \/ B) == A .
You may use any theorems up through 2.4.12 from the list posted on
the course page. I suggest that you start with axiom (10), the lovely
"Golden Rule", to "rewrite" the conjunction -- we have not proved
yet ANY theorems involving conjunction (/\), and it is the ONLY scheme
in the list having anything to do with /\ appearing before 2.4.17'.
Give a bullet proof of 2.4.23(i) in the list of theorem schemes
posted on our course page two weeks ago or so. You may use any of the
11 axiom schemes in your proof, but no other theorem scheme. If you
find that you must use a certain axiom scheme twice in succession,
write out two steps in the proof for it -- use it once in one step,
and a second time in the next step (for full marks).
Of course, you can always spend hours combing through the textbook
looking for similar proofs or these very proofs, or run to the
tutorial leader and try to get him to do a proof for you, but, as
preparation for our upcoming test, I suggest trying instead to figure
out proofs on your own. You will have to do it during the test....
As I said twice in class Tuesday, and it bears repeating,
THE ONLY WAY TO BECOME ADEPT ENOUGH AT PROOFS IN THIS COURSE
TO DO WELL ON TESTS AND EXAM IS TO DO LOTS AND LOTS AND LOTS
OF THEM FOR PRACTICE.
Nothing prevents you, as I also already said, from proving LOTS of
the theorems on our theorem list, for instance. Prove 2.4.11 using
only axioms and lower-numbered theorems. Do the same with 2.4.12
and 2.4.13 and ... and 2.4.26. This is how the instructors become
proficient with proofs -- practice. I know, from my own experience
and from talking with about six of them over the years.
Other proofs may be found as exercises in GT, e.g., 2.4.22 on p. 74,
or the "additional" guys on pp. 86-87. (The above two are perhaps
the first two on these last pages.) Any exercise you give
yourself is good for both your circulation and your state of mind.
And probably for your 1090 grade, if it is exercise at proving
theorems. In your position, I personally would do THIRTY bullet
proofs as practice before Test 1.