1 00:00:00,000 --> 00:00:03,200 Direct proof, so this is the next 
thing that we talk about this week. 2 00:00:03,200 --> 00:00:06,733 So we can prove equalities or inequalities by direct proofs. 3 00:00:06,733 --> 00:00:09,466 Here's one example that I 
did in class: “Prove that 4 00:00:09,466 --> 00:00:12,733 sine of 3 theta is equal to 
3 sine theta plus 4..." oh 5 00:00:12,733 --> 00:00:16,000 oh this should be a minus, I apologize. 
I’ll correct that on the thing. 6 00:00:16,000 --> 00:00:20,433 So it should be sine two theta is equal 
to 3 sine theta minus 4 sine cubed theta. 7 00:00:20,433 --> 00:00:23,700 Now, how do we go about proving this? 
Well we start with the left hand side, 8 00:00:23,700 --> 00:00:25,766 which is sine 3 theta 9 00:00:25,766 --> 00:00:28,866 split it up to 2 theta and theta, 
and then use our trig rules. 10 00:00:28,866 --> 00:00:30,833 Now whenever you use a 
trig rule, something like this, 11 00:00:30,833 --> 00:00:33,100 you should cite what 
you're doing, okay? 12 00:00:33,100 --> 00:00:36,100 For the purposes of this video 
I've just called it a “trig identity”, 13 00:00:36,100 --> 00:00:39,433 for the purposes of Math 
135, this is probably okay. 14 00:00:39,433 --> 00:00:42,400 I think people understand 
what happened here. 15 00:00:42,400 --> 00:00:45,333 Even beyond Math 135, 
this is definitely okay. 16 00:00:45,333 --> 00:00:49,033 I used a trig identity here, and then from 
here to here you actually use two trig identities. 17 00:00:49,033 --> 00:00:52,000 You use a trig identity on sine 2 theta, 
and a trig identity on cos 2 theta, 18 00:00:52,000 --> 00:00:53,833 which is perfectly fine. 19 00:00:53,833 --> 00:00:56,900 You do a little bit of 
simplifying and distributing, 20 00:00:56,900 --> 00:01:00,733 so collect like 
terms, etc, etc. 21 00:01:00,733 --> 00:01:04,233 Then here we use 
the Pythagorean identity, 22 00:01:04,233 --> 00:01:07,200 because cos squared theta is equal to 
1 minus sine squared theta, expand it out, 23 00:01:07,200 --> 00:01:10,500 we get the result that we want. Well the correct result that we want. 24 00:01:10,500 --> 00:01:13,766 So here's the direct proof. Start 
with half of it, and derive 25 00:01:13,766 --> 00:01:16,066 using things that you already know are true, axioms, 26 00:01:16,066 --> 00:01:18,133 to get new truths. Okay? 27 00:01:18,133 --> 00:01:21,066 For the purposes of this course, we 
can take trig identities for granted. 28 00:01:21,066 --> 00:01:24,400 If you ever need them on an exam, we 
will give them to you, in all likelihood, 29 00:01:24,400 --> 00:01:27,466 with the exception of the Pythagorean identity. This should be ingrained into your head. 30 00:01:27,466 --> 00:01:30,333 You might want to tattoo this 
on your arm, it's a possibility… 31 00:01:30,333 --> 00:01:32,066 okay don't tattoo things on your arm… 32 00:01:32,066 --> 00:01:34,433 but, you know , you should know this one. 33 00:01:34,433 --> 00:01:36,866 That's sort of the summary 
that I need to say. 34 00:01:36,866 --> 00:01:40,000 Okay next one. Direct proof 
from a true statement. So 35 00:01:40,000 --> 00:01:41,633 here's another example 
that we saw: “Prove that 36 00:01:41,633 --> 00:01:44,066 5x squared y, minus 3y squared 
is less than or equal to 37 00:01:44,066 --> 00:01:46,833 x to the power 4, plus x 
squared y, plus y squared, 38 00:01:46,833 --> 00:01:49,266 for x and y real numbers.” Okay? 39 00:01:49,266 --> 00:01:52,666 Now the way we proved this in class, I 
kind of just ripped it like a band-aid, right? 40 00:01:52,666 --> 00:01:55,566 So here we have this statement, 
“Since 0 is less than or equal to 41 00:01:55,566 --> 00:01:57,466 x squared minus 
2y all squared…” 42 00:01:57,466 --> 00:02:00,333 remember this is the square of a real number, it's always greater than or equal to 0, 43 00:02:00,333 --> 00:02:04,000 “…we have the following.” So I 
write that down expand it out, 44 00:02:04,000 --> 00:02:08,433 add 5x squared y minus 
3y squared to both sides, 45 00:02:08,433 --> 00:02:11,600 and then simplify. 
We get the results. 46 00:02:11,600 --> 00:02:14,500 Now on the 
one hand, 47 00:02:14,500 --> 00:02:17,933 you can read the proof, and you can follow 
it, and it's reasonably clear what's happening, 48 00:02:17,933 --> 00:02:20,066 right? We haven't done too 
too much that was tricky. 49 00:02:20,066 --> 00:02:22,666 The only thing that 
was maybe clever 50 00:02:22,666 --> 00:02:27,233 as adding 5x squared y minus 3y squared to both sides, but in hindsight, 51 00:02:27,233 --> 00:02:29,766 if you think about it, right, if you're trying to prove this statement up here 52 00:02:29,766 --> 00:02:33,866 then you probably want 5x squared y minus 
3y squared to appear on the left hand side. 53 00:02:33,866 --> 00:02:36,000 So you probably want to 
add that to both sides. 54 00:02:36,000 --> 00:02:38,266 Well it's actually a very “follow 
your nose” kind of proof, 55 00:02:38,266 --> 00:02:40,000 once you have 
the first line. 56 00:02:40,000 --> 00:02:41,800 Now the question is, where does 
this first line come from? 57 00:02:41,800 --> 00:02:43,533 That's where the discovery happens. 58 00:02:43,533 --> 00:02:45,700 These are what I call the “napkin 
computations” and things you do at 59 00:02:45,700 --> 00:02:48,000 Mel's Diner, the things 
you do at the C & D, 60 00:02:48,000 --> 00:02:50,500 something like 
that, right? 61 00:02:50,500 --> 00:02:53,833 Where does it come from? 
Well you take the conclusion, 62 00:02:53,833 --> 00:02:57,366 which is not something you can normally do 
take the conclusion and mess around with it, 63 00:02:57,366 --> 00:03:00,766 but for the purposes of discovery 
you can do whatever you want, right? 64 00:03:00,766 --> 00:03:02,866 I mean if I'm just 
gonna play around, 65 00:03:02,866 --> 00:03:05,700 I'm gonna play around with a 
way that will hopefully help me. 66 00:03:05,700 --> 00:03:08,766 So, the discovery 
of this: well 67 00:03:08,766 --> 00:03:12,000 so I have the 5x squared 
y minus 3y squared 68 00:03:12,000 --> 00:03:12,700 69 00:03:12,700 --> 00:03:14,633 is less than or equal to 
the thing that I want. 70 00:03:14,633 --> 00:03:17,800 I'm gonna bring the terms 
over, which is what I do here, 71 00:03:17,800 --> 00:03:20,033 I'm gonna simplify and 
then I'm gonna notice that, 72 00:03:20,033 --> 00:03:23,066 oh well I can factor this, and I get x squared minus 2y all squared. 73 00:03:23,066 --> 00:03:25,400 And I know that this 
last statement is true. 74 00:03:25,400 --> 00:03:28,000 So as long as I can go 
backwards with all these steps, 75 00:03:28,000 --> 00:03:29,933 then I'm perfectly okay 
with writing the proof, 76 00:03:29,933 --> 00:03:31,933 and it turns out that I can go 
backwards in all these steps. 77 00:03:31,933 --> 00:03:34,900 We'll see in 
a bit that 78 00:03:34,900 --> 00:03:38,666 you can do "if and only if" proofs like 
this, which we'll get to next week. 79 00:03:38,666 --> 00:03:41,600 Something to look forward - 
something to look towards… 80 00:03:41,600 --> 00:03:44,966 well maybe you look forward to 
Math 135 that's okay if you don’t. 81 00:03:44,966 --> 00:03:47,833 That's something 
to look towards, 82 00:03:47,833 --> 00:03:50,866 that you can write these 
proofs like this as long as 83 00:03:50,866 --> 00:03:54,233 all the implications 
are both directions. 84 00:03:54,233 --> 00:03:57,933 85 00:03:57,933 --> 00:04:00,500 Okay. So its a little bit brighter… 
since the sun's gone out… 86 00:04:00,500 --> 00:04:03,200 so direct proof, 
breaking into cases 87 00:04:03,200 --> 00:04:05,366 88 00:04:05,366 --> 00:04:07,133 So I'm gonna just 
turn on the light 89 00:04:07,133 --> 00:04:09,700 just to… 90 00:04:09,700 --> 00:04:12,266 I'll turn on the light to 
make it a little bit clearer. 91 00:04:12,266 --> 00:04:14,700 Hopefully that's okay. Direct 
proof breaking into cases: 92 00:04:14,700 --> 00:04:18,366 “If 2 to the 2n is an odd integer, then 2 
to the minus 2n is also an odd integer.” 93 00:04:18,366 --> 00:04:21,000 Now this is an implication, 
an “if…then”, right? 94 00:04:21,000 --> 00:04:22,966 So remember, implications 
are true if the 95 00:04:22,966 --> 00:04:24,866 hypothesis is true and the conclusion is true, 96 00:04:24,866 --> 00:04:26,600 or the hypothesis is false. 97 00:04:26,600 --> 00:04:29,933 If the hypothesis is false then 
the implication is always true. 98 00:04:29,933 --> 00:04:31,866 So the question now becomes… 
well let's think about this. 99 00:04:31,866 --> 00:04:34,800 2 to the 2n, right, that looks like it 
should be even almost all the time. 100 00:04:34,800 --> 00:04:38,000 Oh here I didn't specify what 
n is, n is just an integer, okay? 101 00:04:38,000 --> 00:04:40,766 102 00:04:40,766 --> 00:04:43,000 So, sorry about that, 
n should be an integer. 103 00:04:43,000 --> 00:04:45,900 But if you think about it, 2 to the 2n is an odd integer, 104 00:04:45,900 --> 00:04:47,600 what does that mean for n? Well 105 00:04:47,600 --> 00:04:50,633 n must…well I mean, in theory, n must be an integer, but 106 00:04:50,633 --> 00:04:53,433 in particular, 
if n is negative, 107 00:04:53,433 --> 00:04:56,333 if n is an integer and n is negative, then 
to the 2 to the 2n isn't even an integer. 108 00:04:56,333 --> 00:04:58,200 It's gonna be 1 over 2 to the 2n, 109 00:04:58,200 --> 00:04:59,933 and that's not 
an integer. 110 00:04:59,933 --> 00:05:04,200 If n is positive, then 2 to the 2n that's gonna be an even integer 111 00:05:04,200 --> 00:05:06,600 because, well 2 will 
divide 2 to the 2n, right? 112 00:05:06,600 --> 00:05:10,133 I can write 2 to the 2n as 
2 times 2 to the 2n minus 1, 113 00:05:10,133 --> 00:05:12,700 and 2n minus 1 is positive when n is positive, 114 00:05:12,700 --> 00:05:15,066 so we see that 2 to the 2n is even. 115 00:05:15,066 --> 00:05:17,666 Thus, what's left? 
Well n must be 0, 116 00:05:17,666 --> 00:05:20,133 and hence 2 to the 0 is equal to 1, which is equal 117 00:05:20,133 --> 00:05:23,933 to 2 to the minus 2n thus, 2 to 
the minus 2n is an odd integer. 118 00:05:23,933 --> 00:05:26,266 So here's an example where we broke into cases. 119 00:05:26,266 --> 00:05:29,133 There are other ways to do 
this so...one way to do it. 120 00:05:29,133 --> 00:05:31,733 So again, we saw positive versus negative 
versus zero, we saw the second one first, 121 00:05:31,733 --> 00:05:34,466 even versus odd, it's another common 
way to break things down. 122 00:05:34,466 --> 00:05:36,700 Sometimes if you 
have integers 123 00:05:36,700 --> 00:05:38,633 and you want to write maybe one is less than or equal to the other 124 00:05:38,633 --> 00:05:40,966 and the other is less than equal to 
the other one, that's possible too. 125 00:05:40,966 --> 00:05:43,433 Lots of ways to break in the cases. I just 
want to give you a couple of examples 126 00:05:43,433 --> 00:05:46,433 of ways to do that. Just make sure that 
your cases cover everything, okay? 127 00:05:46,433 --> 00:05:50,300 When we head into sets next week, we'll 
see a couple more ways to break into cases. 128 00:05:50,300 --> 00:05:50,333