Hello everyone. So I'd like to explain this new video series that I'm going to do.
I’m gonna call it Carmen’s Core Concepts, for
alliteration reasons.
The idea behind these videos is,
not to be a replacement of lectures, but rather to be
a quick summary of what we did this week;
what I felt were the highlights, what were the most important things
and how you can maybe use these techniques to help you in Math 135.
So I've started with week 1.
Again I would love your feedback, please either send me an email
cbruni@uwaterloo.ca
or post on Piazza if I'm teaching this term.
Either way is great, I'd love to hear feedback,
if these are helpful or not.
Anyways I'm gonna begin with week 1,
and the way these are gonna work is I'm going to have...
a table of contents here.
Roughly every bullet will be approximately one minute.
So clearly this is gonna be very very quick.
It's not, you know, you're meant to pause it, and go back, and review it,
it's not meant to replace lectures. It's meant to be a complement to the lectures to say,
“Oh okay, these are the important things we did this week
and these are what I should be focusing on
in order to to develop mastery of mathematics.”
So we're gonna start off with, “What is Math 135?”,
what we cover in it, what's the importance of it.
We’ll talk about truth tables as a definition.
So we talked about truth tables this week and we’d like to
talk about how we can use them to define things.
We'll talk about DeMorgan’s Law and how
truth tables can also be used to prove things.
We talked about chains of equivalences, so using
logical equivalences to show that
statements are equivalent to each other.
Then we talked about direct proofs,
so we just talked about a normal direct proof,
we talked about direct proofs from true statements,
we talked about direct proofs when you break into cases,
and then we finished off the week with the divisibilities and bounds by divisibility.
I should mention that
at this point, before we continue,
don't be alarmed if you didn't cover all of these things.
You'll eventually cover all of these topics, that's perfectly fine.
Some instructors might go in different orders, some instructors
might not have gotten to everything this week, not a big deal.
It's definitely not a big deal; don't take this as like the
gospel to teaching Math 135,
or what you should have done this week, or what you didn't do this week.
Again, these are the topics that I've covered this week,
and so that's what I based this lecture series on.
Okay.
So let's start Math 135. So
Math 135, that's your first proofs course, right?
Remember that proofs differentiate mathematics from science.
Science proof is kind of like, “Well I took a ball and dropped it a hundred times
so if I drop it the next time it's gonna fall.”
That's kind of like proof in science, right, whereas mathematics that's not enough.
Mathematics starts with a set of axioms
and develops a huge network of theorems, lemmas,
propositions, all kinds of things.
You derive new truths from given truths.
We spoke a little bit this week about reading, writing, and discovering proofs.
So there's a difference between these three acts, right?
Reading a proof is very different than reading a normal book, right?
Reading a math textbook should take you a lot longer
than reading a novel.
A novel is written in English, its very easy to understand.
Reading a math textbook’s gonna require a little bit of thought,
a little bit of writing on the side. You have to kind of digest what's going on, things like that.
Writing a proof…
writing a proof is usually not the hardest part of this process.
If I'm discovering - for me anyways - discovering is a lot harder than writing.
But even writing is tricky, right? You need to make sure that you have
the audience level aimed correctly,
and that you're not missing any steps or important
key facts in the proof.
Discovering from a proof, like I said, is always the difficult part.
I always love analogies with napkins.
I used to go to the C&D or to Mel's Diner,
two of my favorite spots to do mathematics,
and you know I used to take their napkins and placemats and doodle on them, right?
That's what I would call the discovering proof stage, these napkin computations.
We’ll see an example of that in a minute.
And sort of the analogy to go through all this is the “Goldilocks analogy”.
You want to write just the right amount in your proof.
You don't want to write too much where people aren't gonna read it,
it's too long, it's too verbose,
you don't want to write too little where you're missing details and important facts.
You want to write just the right amount with good and proper explanation.
You want to make sure that you're using all the correct information here.
Okay so, truth tables as a definition.
So, truth tables are the building blocks of mathematics, right,
I mean this is how... we start off with
some axioms and we develop from here.
So here, we're going to define truth and…
well true and false which we know, right? Like a toggle.
So throughout these explanations of truth tables: let A and B be statements.
What did we see this week? We saw
not’ A, A ‘and’ B, A 'or’ B, or A implies B.
These were the four major ones.
‘Not’ A: Remember if A is true, ‘not’ A is false,
if A is false, ‘not’ A is true.
For the other ones, we have this truth table here.
So here, we're using it as a definition.
So here, we've talked about all four possibilities for A and B.
They're either: true true, true false, false true, or false false.
Again, you don't need to put them in this order, but
it's usually easy to read when they're in a nice order.
Here we have A ‘and’ B so it's true only when
both A and B are true.
We have A ‘or’ B which is false only when both are false
and true otherwise,
and you have A implies B. Now remember this one was the trickier one.
True and true so if A is true and B is true,
then the implication is true. If A is true and B is false,
then the implication is false,
and if A is false then the implication is true, doesn't matter what B is.
So keep that in mind, that implication -
remember the only way it's false is if A is true B is false.
A is false the implication is true.
“If 5 is even then 5 is odd”, right, that's a true statement,
because 5 is not even, therefore
the hypothesis is false and the implication is true.
Keep in mind, A is the hypothesis and B is the conclusion.
That's what we talked about with truth tables as a definition.
We saw DeMorgan's Law, so
the other part of truth tables is that we can actually use them as a method of proof.
So what are DeMorgan's Laws?
Well it's basically a distributive property with A ‘and’ B or A ‘or’ B.
So ‘not’ A ‘and’ B,
the way I kinda remember it is you bring the little ‘not’ inside the A
you change the ‘and’ to ‘or’, or ‘or’ to ‘and’,
and you bring the ‘not’ inside B.
Again, that's sort of the way I remember it.
It's like a distributive law over all of the symbols.
I could prove this by a truth table, right,
so here I’m proving the second one.
So here we have A, B, and then we write A ‘or’ B and then we write ‘not’ A ‘or’ B,
so then you go through, and then we have ‘not’ A and ‘not’ B,
and then you have ‘not’ A ‘and’ ‘not’ B.
Remember - oh I didn't write it here -
So you should always write a concluding sentence, right?
So because the fourth column and the seventh columns
have all the truth values for all possibilities of A and B,
it means that these two statements are logically equivalent.
So you should include that final sentence, I apologize that I didn’t, but that is important.
You want to make sure that the reader knows that you understand why
this table actually means you proved something.
This is a little something subtle there.
DeMorgan's Laws, they help to form chains of equivalences,
which we'll see in a minute.
we're gonna negate implication next, and we'll see how we use
DeMorgan's Laws and other chains of equivalences to derive new truths.
So one way to derive new truth is using the truth table and the other way,
with these statements, is to use a chain of equivalence.
So “Using Chains of Equivalences”:
recall one thing that we did this week was A implies B is equivalent to ‘not’ A ‘or’ B, right?
Either the hypothesis is false, or the conclusion is true.
If one of those two things are true then the implication is true,
that’s one way to remember it.
So prove that the negation of an implication is A ‘and’ ‘not’ B.
So how do we do that? Well here
we can use the proposition that we proved in class by a truth table.
We can throw that in so by the above proposition on this page, we have this.
Then we have ‘not’ ‘not’ A ‘and’ B. that's DeMorgan's Law, right,
we bring the ‘not’ into everything.
So ‘not’ ‘not’ A, ‘not’ of ‘or’ I like to think of as ‘and’,
and ‘not’ B is ‘not’ B.
Then ‘not’ ‘not’ A that’s just A
from a proposition from class that double negation should cancel.
And that's it. So here's an example where we can use
equivalent statements to prove new truths in mathematics.
That was sort of our brief
look at logic for Math 135.