Direct proof, so this is the next thing that we talk about this week.
So we can prove equalities or inequalities by direct proofs.
Here's one example that I did in class: “Prove that
sine of 3 theta is equal to 3 sine theta plus 4..." oh
oh this should be a minus, I apologize. I’ll correct that on the thing.
So it should be sine two theta is equal to 3 sine theta minus 4 sine cubed theta.
Now, how do we go about proving this? Well we start with the left hand side,
which is sine 3 theta
split it up to 2 theta and theta, and then use our trig rules.
Now whenever you use a trig rule, something like this,
you should cite what you're doing, okay?
For the purposes of this video I've just called it a “trig identity”,
for the purposes of Math 135, this is probably okay.
I think people understand what happened here.
Even beyond Math 135, this is definitely okay.
I used a trig identity here, and then from here to here you actually use two trig identities.
You use a trig identity on sine 2 theta, and a trig identity on cos 2 theta,
which is perfectly fine.
You do a little bit of simplifying and distributing,
so collect like terms, etc, etc.
Then here we use the Pythagorean identity,
because cos squared theta is equal to 1 minus sine squared theta, expand it out,
we get the result that we want. Well the correct result that we want.
So here's the direct proof. Start with half of it, and derive
using things that you already know are true, axioms,
to get new truths. Okay?
For the purposes of this course, we can take trig identities for granted.
If you ever need them on an exam, we will give them to you, in all likelihood,
with the exception of the Pythagorean identity. This should be ingrained into your head.
You might want to tattoo this on your arm, it's a possibility…
okay don't tattoo things on your arm…
but, you know , you should know this one.
That's sort of the summary that I need to say.
Okay next one. Direct proof from a true statement. So
here's another example that we saw: “Prove that
5x squared y, minus 3y squared is less than or equal to
x to the power 4, plus x squared y, plus y squared,
for x and y real numbers.” Okay?
Now the way we proved this in class, I kind of just ripped it like a band-aid, right?
So here we have this statement, “Since 0 is less than or equal to
x squared minus 2y all squared…”
remember this is the square of a real number, it's always greater than or equal to 0,
“…we have the following.” So I write that down expand it out,
add 5x squared y minus 3y squared to both sides,
and then simplify. We get the results.
Now on the one hand,
you can read the proof, and you can follow it, and it's reasonably clear what's happening,
right? We haven't done too too much that was tricky.
The only thing that was maybe clever
as adding 5x squared y minus 3y squared to both sides, but in hindsight,
if you think about it, right, if you're trying to prove this statement up here
then you probably want 5x squared y minus 3y squared to appear on the left hand side.
So you probably want to add that to both sides.
Well it's actually a very “follow your nose” kind of proof,
once you have the first line.
Now the question is, where does this first line come from?
That's where the discovery happens.
These are what I call the “napkin computations” and things you do at
Mel's Diner, the things you do at the C & D,
something like that, right?
Where does it come from? Well you take the conclusion,
which is not something you can normally do take the conclusion and mess around with it,
but for the purposes of discovery you can do whatever you want, right?
I mean if I'm just gonna play around,
I'm gonna play around with a way that will hopefully help me.
So, the discovery of this: well
so I have the 5x squared y minus 3y squared
is less than or equal to the thing that I want.
I'm gonna bring the terms over, which is what I do here,
I'm gonna simplify and then I'm gonna notice that,
oh well I can factor this, and I get x squared minus 2y all squared.
And I know that this last statement is true.
So as long as I can go backwards with all these steps,
then I'm perfectly okay with writing the proof,
and it turns out that I can go backwards in all these steps.
We'll see in a bit that
you can do "if and only if" proofs like this, which we'll get to next week.
Something to look forward - something to look towards…
well maybe you look forward to Math 135 that's okay if you don’t.
That's something to look towards,
that you can write these proofs like this as long as
all the implications are both directions.
Okay. So its a little bit brighter… since the sun's gone out…
so direct proof, breaking into cases
So I'm gonna just turn on the light
just to…
I'll turn on the light to make it a little bit clearer.
Hopefully that's okay. Direct proof breaking into cases:
“If 2 to the 2n is an odd integer, then 2 to the minus 2n is also an odd integer.”
Now this is an implication, an “if…then”, right?
So remember, implications are true if the
hypothesis is true and the conclusion is true,
or the hypothesis is false.
If the hypothesis is false then the implication is always true.
So the question now becomes… well let's think about this.
2 to the 2n, right, that looks like it should be even almost all the time.
Oh here I didn't specify what n is, n is just an integer, okay?
So, sorry about that, n should be an integer.
But if you think about it, 2 to the 2n is an odd integer,
what does that mean for n? Well
n must…well I mean, in theory, n must be an integer, but
in particular, if n is negative,
if n is an integer and n is negative, then to the 2 to the 2n isn't even an integer.
It's gonna be 1 over 2 to the 2n,
and that's not an integer.
If n is positive, then 2 to the 2n that's gonna be an even integer
because, well 2 will divide 2 to the 2n, right?
I can write 2 to the 2n as 2 times 2 to the 2n minus 1,
and 2n minus 1 is positive when n is positive,
so we see that 2 to the 2n is even.
Thus, what's left? Well n must be 0,
and hence 2 to the 0 is equal to 1, which is equal
to 2 to the minus 2n thus, 2 to the minus 2n is an odd integer.
So here's an example where we broke into cases.
There are other ways to do this so...one way to do it.
So again, we saw positive versus negative versus zero, we saw the second one first,
even versus odd, it's another common way to break things down.
Sometimes if you have integers
and you want to write maybe one is less than or equal to the other
and the other is less than equal to the other one, that's possible too.
Lots of ways to break in the cases. I just want to give you a couple of examples
of ways to do that. Just make sure that your cases cover everything, okay?
When we head into sets next week, we'll see a couple more ways to break into cases.