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Hello everyone. So I'd like to explain
this new video series that I'm going to
do.

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I’m gonna call it Carmen’s Core
Concepts, for

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alliteration reasons.

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The idea behind 
these videos is,

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not to be a replacement of 
lectures, but rather to be

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a quick summary of 
what we did this week;

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what I felt were the highlights, what 
were the most important things

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and how you can maybe use these 
techniques to help you in Math 135.

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So I've started with week 1.

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Again I would love your feedback, 
please either send me an email

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cbruni@uwaterloo.ca

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or post on Piazza if
I'm teaching this term.

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Either way is great, I'd 
love to hear feedback,

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if these are helpful or not.

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Anyways I'm gonna 
begin with week 1,

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and the way these are gonna 
work is I'm going to have...

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a table of
contents here.

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Roughly every bullet will be 
approximately one minute.

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So clearly this is gonna 
be very very quick.

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It's not, you know, you're meant
to pause it, and go back, and review it,

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it's not meant to replace lectures. It's 
meant to be a complement to the 
lectures to say,

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“Oh okay, these are the important 
things we did this week

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and these are what I
should be focusing on

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in order to to develop 
mastery of mathematics.”

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So we're gonna start off 
with, “What is Math 135?”,

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what we cover in it, what's the
importance of it.

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We’ll talk about truth
tables as a definition.

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So we talked about truth tables 
this week and we’d like to

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talk about how we can 
use them to define things.

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We'll talk about DeMorgan’s 
Law and how

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truth tables can also be
used to prove things.

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We talked about chains of
equivalences, so using

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logical equivalences 
to show that

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statements are
equivalent to each other.

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Then we talked 
about direct proofs, 

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so we just talked about 
a normal direct proof,

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we talked about direct 
proofs from true statements,

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we talked about direct proofs
when you break into cases,

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and then we finished off the week 
with the divisibilities and bounds by
divisibility.

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I should mention that

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at this point, before 
we continue,

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don't be alarmed if you didn't 
cover all of these things.

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You'll eventually cover all of
these topics, that's perfectly fine.

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Some instructors might go in 
different orders, some instructors

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might not have gotten to everything 
this week, not a big deal.

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It's definitely not a big deal; 
don't take this as like the

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gospel to teaching 
Math 135,

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or what you should have done this 
week, or what you didn't do this week.

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Again, these are the topics 
that I've covered this week,

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and so that's what I based 
this lecture series on.

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Okay.

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So let's start 
Math 135. So

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Math 135, that's your 
first proofs course, right?

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Remember that proofs differentiate 
mathematics from science.

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Science proof is kind of like, “Well 
I took a ball and dropped it a hundred times

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so if I drop it the next
time it's gonna fall.”

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That's kind of like proof in science, right, 
whereas mathematics that's not enough.

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Mathematics starts 
with a set of axioms

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and develops a huge network 
of theorems, lemmas, 

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propositions, all kinds of things.

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You derive new truths 
from given truths.

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We spoke a little bit this week about 
reading, writing, and discovering proofs.

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So there's a difference
between these three acts, right?

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Reading a proof is very different 
than reading a normal book, right?

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Reading a math textbook 
should take you a lot longer

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than reading a novel. 

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A novel is written in English,
its very easy to understand.

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Reading a math textbook’s gonna 
require a little bit of thought,

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a little bit of writing on the side. You 
have to kind of digest what's going 
on, things like that.

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Writing a proof…

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writing a proof is usually not the 
hardest part of this process.

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If I'm discovering - for me anyways - 
discovering is a lot harder than writing.

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But even writing is tricky, right? You
need to make sure that you have

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the audience level 
aimed correctly,

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and that you're not missing 
any steps or important

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key facts in the proof.

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Discovering from a proof, like I 
said, is always the difficult part.

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I always love analogies
with napkins.

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I used to go to the 
C&D or to Mel's Diner,

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two of my favorite spots 
to do mathematics,

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and you know I used to take their napkins
and placemats and doodle on them, right?

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That's what I would call the discovering 
proof stage, these napkin computations.

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We’ll see an example 
of that in a minute.

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And sort of the analogy to go through 
all this is the “Goldilocks analogy”.

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You want to write just the
right amount in your proof.

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You don't want to write too much 
where people aren't gonna read it, 

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it's too long, it's too verbose,

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you don't want to write too little where 
you're missing details and important facts.

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You want to write just the right amount 
with good and proper explanation.

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You want to make sure that you're 
using all the correct information here.

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Okay so, truth tables 
as a definition.

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So, truth tables are the building 
blocks of mathematics, right,

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I mean this is how...
we start off with

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some axioms and we
develop from here.

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So here, we're going 
to define truth and…

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well true and false which we 
know, right? Like a toggle.

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So throughout these explanations of 
truth tables: let A and B be statements.

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What did we see 
this week? We saw

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not’ A, A ‘and’ B, A 
'or’ B, or A implies B.

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These were the 
four major ones.

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‘Not’ A: Remember if A 
is true, ‘not’ A is false,

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if A is false, 
‘not’ A is true.

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For the other ones, we 
have this truth table here.

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So here, we're using 
it as a definition.

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So here, we've talked about 
all four possibilities for A and B.

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They're either: true true, true false,
false true, or false false.

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Again, you don't need to 
put them in this order, but

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it's usually easy to read when
they're in a nice order.

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Here we have A ‘and’ B 
so it's true only when

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both A and 
B are true.

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We have A ‘or’ B which is
false only when both are false

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and true
otherwise,

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and you have A implies B. Now 
remember this one was the trickier one.

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True and true so if A is 
true and B is true,

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then the implication is true. 
If A is true and B is false,

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then the implication is false,

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and if A is false then the implication
is true, doesn't matter what B is.

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So keep that in 
mind, that implication -

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remember the only way it's 
false is if A is true B is false.

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A is false the
implication is true.

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“If 5 is even then 5 is odd”, 
right, that's a true statement,

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because 5 is not 
even, therefore

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the hypothesis is false 
and the implication is true.

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Keep in mind, A is the hypothesis 
and B is the conclusion.

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That's what we talked about 
with truth tables as a definition.

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We saw DeMorgan's Law, so

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the other part of truth tables is that we can 
actually use them as a method of proof.

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So what are 
DeMorgan's Laws?

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Well it's basically a distributive 
property with A ‘and’ B or A ‘or’ B.

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So ‘not’ A ‘and’ B,

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the way I kinda remember it is you 
bring the little ‘not’ inside the A

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you change the ‘and’ to 
‘or’, or ‘or’ to ‘and’,

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and you bring the 
‘not’ inside B.

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Again, that's sort of 
the way I remember it.

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It's like a distributive law
over all of the symbols.

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I could prove this 
by a truth table, right,

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so here I’m proving 
the second one.

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So here we have A, B, and then we 
write A ‘or’ B and then we write ‘not’ A ‘or’ B,

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so then you go through, and then 
we have ‘not’ A and ‘not’ B,

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and then you have 
‘not’ A ‘and’ ‘not’ B. 

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Remember - oh I didn't write it here -

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So you should always write a
concluding sentence, right?

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So because the fourth column 
and the seventh columns 

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have all the truth values for 
all possibilities of A and B,

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it means that these two statements
are logically equivalent.

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So you should include that final sentence, 
I apologize that I didn’t, but that is important.

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You want to make sure that the reader
knows that you understand why

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this table actually means 
you proved something.

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This is a little 
something subtle there.

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DeMorgan's Laws, they help to 
form chains of equivalences,

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which we'll see 
in a minute.

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we're gonna negate implication 
next, and we'll see how we use

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DeMorgan's Laws and other chains of
equivalences to derive new truths.

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So one way to derive new truth is
using the truth table and the other way,

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with these statements, is to 
use a chain of equivalence.

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So “Using Chains of Equivalences”:

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recall one thing that we did this week was 
A implies B is equivalent to ‘not’ A ‘or’ B, right?

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Either the hypothesis is
false, or the conclusion is true.

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If one of those two things are true 
then the implication is true,

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that’s one way to
remember it.

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So prove that the negation of 
an implication is A ‘and’ ‘not’ B.

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So how do we 
do that? Well here

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we can use the proposition that we 
proved in class by a truth table.

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We can throw that in so by the above 
proposition on this page, we have this.

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Then we have ‘not’ ‘not’ A ‘and’ B.
that's DeMorgan's Law, right,

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we bring the ‘not’ 
into everything.

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So ‘not’ ‘not’ A, ‘not’ of ‘or’ 
I like to think of as ‘and’,

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and ‘not’ B 
is ‘not’ B.

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Then ‘not’ ‘not’ A
that’s just A

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from a proposition from class that
double negation should cancel.

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And that's it. So here's an 
example where we can use

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equivalent statements to prove 
new truths in mathematics.

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That was sort
of our brief

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look at logic 
for Math 135.