Hello everyone. Welcome to week 3 of Carmen's Core Concepts.
Again in these videos, we go over the current week in Math 135
and just give you a little bit of a highlight reel of
what you did this week, and hopefully how…
it's just a summary of what we did this week.
Again reminder that this isn't a substitute for going to lectures,
I'm definitely gonna go a lot faster than in lectures
and only cover really key points.
The other thing of course, this also isn't substitute for actually doing the math,
right? Just like if I watch a swimmer swim,
doesn't make me a good swimmer.
Watching me do math, doesn't make you a good mathematician.
You have to actually go out and do mathematics.
So translating from mathematics to English.
The first thing I'm going to say is going to sound really obvious:
make sure you know what a question is asking before answering it.
I don't mean reading it I mean before answering it.
Well why is that important? Well…
if you're trying to answer a question that you
don't actually know what it says, it's gonna be a lot harder
than if you actually know what the question says.
So it's really important to actually try to make sure that you
really do understand what symbols are being used, what they mean,
and what they are. I mean don't try to answer a question
if you don't understand it, right? We're going to know you don't understand,
and you're not going to be able to answer the problem. So stop,
read the question, really make sure you understand what the symbols mean.
So some key words.
We went through a couple of examples in class this week where we had
“for all”s and “there exists”s
missing from the sentence, okay?
So some key tip-offs to use a “for all” statement
are things like “always”, “whenever”, “for any”, “none”,
“no”.
And keywords for meaning “there exists”, you're kind of looking for
“some”, “has a”, “there is” things like that, okay?
So here's some examples: so no multiple of 15 plus any multiple 6 equals 100.
This is an example of a “for all” statement.
Why? Because “no multiple”… so for every single multiple
of 15, and for every single of 6, if I add them together, I never get 100.
That's the way you should be reading it, and again, this takes practice.
The second one I have is
well here's an example of symbols. So if n is an element of Z
implies that there exists an integer m such that m is greater than n.
What is it really saying? Well if I start with any integer,
then there's always a bigger integer, than it.
So what am I saying? Well there is no greatest integer;
it's another way to say it, okay?
Again, there's lots of ways to translate it,
so don't just translate it line for line.
You should try to get an understanding of what the thing is saying.
If you do that, your life will be easier; it'll be better for you.
It's easier to start a problem when you know what it's asking.
Contrapositive. Okay.
So we have two more proof techniques that we proved this week,
and the first one is a proof by contrapositive.
So what’s the idea behind this? Well if I have a hypothesis and it implies the conclusion,
so H implies C, an implication,
then this is logically equivalent to if I take
the negation of the conclusion and show the negation of the hypothesis.
It's another way to prove things, so instead of starting with a hypothesis
and proving the conclusion, you can start with the negation of the conclusion
and prove the negation of the hypothesis.
What's a proof of this? Well a proof of this,
you can actually do this symbolically. You can use a truth table as well.
So H implies C is equivalent to, we sorted out this in class, ‘not’ H ‘or’ C.
I'm gonna flip the order around and that's still the same: C ‘or’ ‘not’ H.
Instead of C I'm gonna write it as the negation of the negation of C.
Two negations… two wrongs make a right.
And what's this? Well if I
look at this, this is ‘not’ the negation of C implies the negation of H,
well that's the same as ‘not’ C implies ‘not’ H
So for example, an example of when…
so a physical example, let's look at this one:
“7 does not divide n implies that 14 does not divide n.”
The non-existent statements,
these are usually really good times to use the contrapositive.
"7 does not divide n", that's a tough situation to start with,
right, because all you know is that there is no integer such that 7
times that integer is n.
It's tough to deal with.
It's a lot easier to deal with, well 14 divides n
well I know what that is. There is an integer [k]
such that 14 times k equals n,
and then try to show that 7 divides n.
So when you have these ‘not’ statements... specifically in the conclusion but
in the hypothesis and/or the conclusion,
it might be a good idea to try to the contrapositive.
Just a general rule of thumb. Doesn't always work
but at least it gives you something to try.
Let's see an example.
So here's an example where it's maybe not so obvious to use but
it will become obvious in a little bit,
“Suppose that we have two real numbers a and b
and that the product a times b is in
the set of real numbers
set differenced with the set of rational numbers.”
So what does that mean? Well we're taking the real numbers,
and we're removing the rational numbers.
Well it's going to leave us with the irrational numbers.
“Show that either a irrational or b is irrational.”
So that's the statement. By the way, this is what I mean by reading the problem, right?
If you don't understand what R minus Q is,
then it's gonna be a lot harder to answer this question,
but if you do understand it's just the set of irrational numbers,
it makes solving the problem a lot easier.
Here I'm showing that something is irrational. What does it mean to be irrational?
If you define something
with the words, “Well you're not rational,” that should
tip off to you hey maybe I should use a contrapositive,
or maybe I should use a proof by contradiction, which we'll see later.
In this case, definitely a proof by contrapositive is the correct way to go.
So we're going to proceed with the contrapositive,
so we're going suppose that… what's the contrapositive of this?
We're going to take the negation of the conclusion.
Well the conclusion is, “a is irrational or b is irrational.”
So what's the negation of that? Ask yourself and say,
“Okay well that's a is rational and b is rational.”
That's a lot easier to deal with.
Dealing with a is rational and b is rational and proving something, that's a lot easier.
So we're gonna start with that.
What does that mean?
Well there exists integers k, l, m, n
such that a equals k over l, and b equals m over n, with l and n not 0.
Then what? Well okay what are we trying to show?
Let’s making sure that we understand what we're trying to show.
So we started with the negation of the conclusion. We're trying to show
the negation of the hypothesis is true.
Well what's the hypothesis? The hypothesis is:
“a b is irrational.” Don't...
worry about the a and b, inside the real numbers. That's
sort of… that's above the statement, right? This is like a
“for all a and b inside the real numbers,
this implies this.” That's the way you should read this.
So here we want to show the negation the hypothesis which is: “a b is rational”.
Well what does it mean for a b to be rational? Well I can write it as
some integer divided by some other integer,
and the bottom integer is not 0.
Well if l and n weren’t 0, the product is definitely not 0.
All of these are products of integers, so they’re still an integer,
therefore a times b is going to be an integer over a non-zero integer,
that’s a rational, and we’re done.
So there's an example of the use of contrapositive.
When the conclusion contains the negation of
an existence sort of thing.
Again, just rules of thumb, when you use it, try it, and see if it works.
If it works then you should use it. If it doesn't work, then try something else.
Again, our toolbox of techniques are getting bigger, okay?
So when you see a new problem,
it's not going to be as obvious anymore what tool to use.
In real life, you know it's very easy to know what screwdriver to use,
or maybe not so obvious what wrench to use, maybe that's a better analogy here.
Figuring out the size of the wrench to use can be tricky.
You have to actually measure and try some things.
But that's the idea now, okay?
So we have a lot of tools. Now we have a wrench with lots of different, you know, socket pieces.