MATH 8, SECTION 1, WEEK 1  RECITATION NOTES
TA: PADRAIC BARTLETT
Abstract.
These are the notes from Wednesday, Sept.
29th’s lecture.
In
this talk, we continue our discussion on proofs by contradiction, and examine
proofs by induction.
1.
Random Question
Question 1.1.
For any
n
in
N
, can you find a way to tile
1
the shape
n
2n
2n
n
n
n
with triominoes made out of three 1x1 squares of the form
?
2.
Administrivia and Announcements
•
Class times will remain at 23 MWF for the quarter; while some other time
slots are marginally better, none of them are sufficiently better to clear a
class change with the Registrar (and running two sections appears to be
an impossibility as well.)
Sorry, CS1 students!
As always, any students
who can’t make Math 8 consistently are welcome to attend as unregistered
students when they can, and read the course notes online.
•
Homework policy! In your Math 1 HW,
–
you
cannot
cite the Math 8 notes or lectures! Your section TA will
have no idea what you’re talking about.
–
you
can
use your Math 8 notes for reference and inspiration!
I.e.
direct copying of the notes = bad, but using them to remember how
a proof goes or some clever trick, and then writing up the results in
your own words = good.
1
A tiling of some region R with some shape S is a way of covering all of the points in R with
translations, rotations, and reflections of the shape S, so that (1) all of the copies of S lie inside
of R, and (2) none of the copies of S overlap (except for possibly on their boundaries.)
1
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TA: PADRAIC BARTLETT
3.
Proofs by Contradiction, Part II
Last time, we described proofs by contradiction as the following process: suppose
we want to prove that some statement
P
is true. How can we do this? Well, there
are only two possiblities:
either
P
is true, or
¬
P
is.
So:
if we show that
¬
P
is
impossible – in other words, that assuming
¬
P
leads to a contradiction – then by
process of elimination we have tht
P
must hold!
On Monday, we gave one example of a proof by contradiction; today, we have
two more proofs to further illustrate the method.
Lemma 3.1.
(Euclid) There are infinitely many prime numbers.
Proof.
(N.b.: for the entertainment of those who’ve already encountered Euclid’s
proof about prime numbers, the proof below is written in the “way of prime” – i.e.
all of the words below contain prime numbers of letters. Silly, yes. But fun!)
Suppose not: in other words, suppose
n
prime numbers exist, for
n
∈
N
. Label
these prime numbers
p
1
, p
2
, . . . p
n
,
and examine the product
p
1
·
p
2
·
. . .
·
p
n
.
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 Fall '09
 Math, Calculus, TA, Natural number, Fibonacci number, Mathematical proof, inductive step

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