1 00:00:00,000 --> 00:00:04,000 Hello everyone. Welcome to week 
3 of Carmen's Core Concepts. 2 00:00:04,000 --> 00:00:07,866 Again in these videos, we go over 
the current week in Math 135 3 00:00:07,866 --> 00:00:11,133 and just give you a little 
bit of a highlight reel of 4 00:00:11,133 --> 00:00:14,066 what you did this week, 
and hopefully how… 5 00:00:14,066 --> 00:00:16,000 it's just a summary of 
what we did this week. 6 00:00:16,000 --> 00:00:19,133 Again reminder that this isn't a 
substitute for going to lectures, 7 00:00:19,133 --> 00:00:21,100 I'm definitely gonna go a lot 
faster than in lectures 8 00:00:21,100 --> 00:00:24,400 and only cover 
really key points. 9 00:00:24,400 --> 00:00:28,133 The other thing of course, this also isn't substitute for actually doing the math, 10 00:00:28,133 --> 00:00:30,700 right? Just like if I watch 
a swimmer swim, 11 00:00:30,700 --> 00:00:32,166 doesn't make me 
a good swimmer. 12 00:00:32,166 --> 00:00:34,333 Watching me do math, doesn't make 
you a good mathematician. 13 00:00:34,333 --> 00:00:36,400 You have to actually go out and do mathematics. 14 00:00:36,400 --> 00:00:38,933 So translating from 
mathematics to English. 15 00:00:38,933 --> 00:00:40,566 The first thing I'm going to say is 
going to sound really obvious: 16 00:00:40,566 --> 00:00:44,400 make sure you know what a question 
is asking before answering it. 17 00:00:44,400 --> 00:00:46,633 I don't mean reading it I mean before answering it. 18 00:00:46,633 --> 00:00:49,066 Well why is that 
important? Well… 19 00:00:49,066 --> 00:00:50,333 20 00:00:50,333 --> 00:00:52,533 if you're trying to answer a question that you 21 00:00:52,533 --> 00:00:54,666 don't actually know what it says, 
it's gonna be a lot harder 22 00:00:54,666 --> 00:00:57,100 than if you actually know 
what the question says. 23 00:00:57,100 --> 00:01:00,400 So it's really important to actually 
try to make sure that you 24 00:01:00,400 --> 00:01:04,233 really do understand what symbols 
are being used, what they mean, 25 00:01:04,233 --> 00:01:06,300 and what they are. I mean 
don't try to answer a question 26 00:01:06,300 --> 00:01:09,233 if you don't understand it, right? We're 
going to know you don't understand, 27 00:01:09,233 --> 00:01:11,900 and you're not going to be able 
to answer the problem. So stop, 28 00:01:11,900 --> 00:01:15,033 read the question, really make sure you understand what the symbols mean. 29 00:01:15,033 --> 00:01:17,266 So some 
key words. 30 00:01:17,266 --> 00:01:20,066 We went through a couple of examples 
in class this week where we had 31 00:01:20,066 --> 00:01:22,233 “for all”s and 
“there exists”s 32 00:01:22,233 --> 00:01:24,400 missing from the 
sentence, okay? 33 00:01:24,400 --> 00:01:28,400 So some key tip-offs to 
use a “for all” statement 34 00:01:28,400 --> 00:01:32,400 are things like “always”, “whenever”, “for any”, “none”, 35 00:01:32,400 --> 00:01:34,400 “no”. 36 00:01:34,400 --> 00:01:36,966 And keywords for meaning “there 
exists”, you're kind of looking for 37 00:01:36,966 --> 00:01:40,733 “some”, “has a”, “there is” 
things like that, okay? 38 00:01:40,733 --> 00:01:41,866 39 00:01:41,866 --> 00:01:45,733 So here's some examples: so no multiple 
of 15 plus any multiple 6 equals 100. 40 00:01:45,733 --> 00:01:48,033 This is an example of a “for all” statement. 41 00:01:48,033 --> 00:01:51,600 Why? Because “no multiple”… 
so for every single multiple 42 00:01:51,600 --> 00:01:55,733 of 15, and for every single of 6, if I 
add them together, I never get 100. 43 00:01:55,733 --> 00:01:59,066 That's the way you should be reading 
it, and again, this takes practice. 44 00:01:59,066 --> 00:02:01,500 The second 
one I have is 45 00:02:01,500 --> 00:02:03,933 well here's an example of symbols. 
So if n is an element of Z 46 00:02:03,933 --> 00:02:07,633 implies that there exists an integer 
m such that m is greater than n. 47 00:02:07,633 --> 00:02:10,600 What is it really saying? Well 
if I start with any integer, 48 00:02:10,600 --> 00:02:12,200 49 00:02:12,200 --> 00:02:14,566 then there's always a 
bigger integer, than it. 50 00:02:14,566 --> 00:02:16,666 So what am I saying? Well 
there is no greatest integer; 51 00:02:16,666 --> 00:02:18,533 it's another way 
to say it, okay? 52 00:02:18,533 --> 00:02:20,566 Again, there's lots of 
ways to translate it, 53 00:02:20,566 --> 00:02:22,366 so don't just translate 
it line for line. 54 00:02:22,366 --> 00:02:24,866 You should try to get an understanding 
of what the thing is saying. 55 00:02:24,866 --> 00:02:28,400 If you do that, your life will be easier; it'll be better for you. 56 00:02:28,400 --> 00:02:32,400 It's easier to start a problem when 
you know what it's asking. 57 00:02:32,400 --> 00:02:32,466 58 00:02:32,466 --> 00:02:34,166 59 00:02:34,166 --> 00:02:36,366 Contrapositive. Okay. 60 00:02:36,366 --> 00:02:40,333 So we have two more proof techniques that we proved this week, 61 00:02:40,333 --> 00:02:42,900 and the first one is 
a proof by contrapositive. 62 00:02:42,900 --> 00:02:46,800 So what’s the idea behind this? Well if I 
have a hypothesis and it implies the conclusion, 63 00:02:46,800 --> 00:02:48,800 so H implies C, an implication, 64 00:02:48,800 --> 00:02:51,000 then this is logically equivalent to if I take 65 00:02:51,000 --> 00:02:55,266 the negation of the conclusion and 
show the negation of the hypothesis. 66 00:02:55,266 --> 00:02:56,300 67 00:02:56,300 --> 00:02:58,900 It's another way to prove things, so 
instead of starting with a hypothesis 68 00:02:58,900 --> 00:03:02,666 and proving the conclusion, you can start 
with the negation of the conclusion 69 00:03:02,666 --> 00:03:05,466 and prove the negation 
of the hypothesis. 70 00:03:05,466 --> 00:03:06,133 71 00:03:06,133 --> 00:03:08,066 What's a proof of this? 
Well a proof of this, 72 00:03:08,066 --> 00:03:11,300 you can actually do this symbolically. You can use a truth table as well. 73 00:03:11,300 --> 00:03:15,733 So H implies C is equivalent to, we 
sorted out this in class, ‘not’ H ‘or’ C. 74 00:03:15,733 --> 00:03:19,300 I'm gonna flip the order around and 
that's still the same: C ‘or’ ‘not’ H. 75 00:03:19,300 --> 00:03:22,466 Instead of C I'm gonna write it as 
the negation of the negation of C. 76 00:03:22,466 --> 00:03:26,133 Two negations… two 
wrongs make a right. 77 00:03:26,133 --> 00:03:27,300 And what's 
this? Well if I 78 00:03:27,300 --> 00:03:31,133 look at this, this is ‘not’ the negation 
of C implies the negation of H, 79 00:03:31,133 --> 00:03:35,300 well that's the same as 
‘not’ C implies ‘not’ H 80 00:03:35,300 --> 00:03:38,333 So for example, an 
example of when… 81 00:03:38,333 --> 00:03:40,466 so a physical example, 
let's look at this one: 82 00:03:40,466 --> 00:03:44,400 “7 does not divide n implies that 14 does not divide n.” 83 00:03:44,400 --> 00:03:46,033 The non-existent 
statements, 84 00:03:46,033 --> 00:03:48,700 these are usually really good 
times to use the contrapositive. 85 00:03:48,700 --> 00:03:51,633 "7 does not divide n", that's a 
tough situation to start with, 86 00:03:51,633 --> 00:03:55,266 right, because all you know is that 
there is no integer such that 7 87 00:03:55,266 --> 00:03:57,266 times that 
integer is n. 88 00:03:57,266 --> 00:03:58,800 It's tough 
to deal with. 89 00:03:58,800 --> 00:04:00,933 It's a lot easier to deal 
with, well 14 divides n 90 00:04:00,933 --> 00:04:02,933 well I know what that 
is. There is an integer [k] 91 00:04:02,933 --> 00:04:04,900 such that 14 
times k equals n, 92 00:04:04,900 --> 00:04:07,300 and then try to show 
that 7 divides n. 93 00:04:07,300 --> 00:04:11,300 So when you have these ‘not’ statements...
specifically in the conclusion but 94 00:04:11,300 --> 00:04:13,266 in the hypothesis 
and/or the conclusion, 95 00:04:13,266 --> 00:04:16,100 it might be a good idea 
to try to the contrapositive. 96 00:04:16,100 --> 00:04:18,866 Just a general rule of 
thumb. Doesn't always work 97 00:04:18,866 --> 00:04:21,200 but at least it gives 
you something to try. 98 00:04:21,200 --> 00:04:22,233 99 00:04:22,233 --> 00:04:23,800 Let's see 
an example. 100 00:04:23,800 --> 00:04:26,733 So here's an example where it's 
maybe not so obvious to use but 101 00:04:26,733 --> 00:04:28,633 it will become obvious 
in a little bit, 102 00:04:28,633 --> 00:04:30,666 “Suppose that we have two 
real numbers a and b 103 00:04:30,666 --> 00:04:33,333 and that the product 
a times b is in 104 00:04:33,333 --> 00:04:35,166 the set of 
real numbers 105 00:04:35,166 --> 00:04:38,266 set differenced with the 
set of rational numbers.” 106 00:04:38,266 --> 00:04:40,033 So what does that mean? Well we're taking the real numbers, 107 00:04:40,033 --> 00:04:41,966 and we're removing 
the rational numbers. 108 00:04:41,966 --> 00:04:44,633 Well it's going to leave us 
with the irrational numbers. 109 00:04:44,633 --> 00:04:47,300 “Show that either a irrational or b is irrational.” 110 00:04:47,300 --> 00:04:48,533 111 00:04:48,533 --> 00:04:51,033 So that's the statement. By the way, this 
is what I mean by reading the problem, right? 112 00:04:51,033 --> 00:04:53,166 If you don't understand 
what R minus Q is, 113 00:04:53,166 --> 00:04:54,866 then it's gonna be a lot harder 
to answer this question, 114 00:04:54,866 --> 00:04:57,066 but if you do understand it's just 
the set of irrational numbers, 115 00:04:57,066 --> 00:05:00,233 it makes solving the 
problem a lot easier. 116 00:05:00,233 --> 00:05:04,133 Here I'm showing that something is irrational. 
What does it mean to be irrational? 117 00:05:04,133 --> 00:05:06,366 If you define 
something 118 00:05:06,366 --> 00:05:09,400 with the words, “Well you're 
not rational,” that should 119 00:05:09,400 --> 00:05:11,733 tip off to you hey maybe I should use a contrapositive, 120 00:05:11,733 --> 00:05:14,366 or maybe I should use a proof by 
contradiction, which we'll see later. 121 00:05:14,366 --> 00:05:17,833 In this case, definitely a proof by 
contrapositive is the correct way to go. 122 00:05:17,833 --> 00:05:19,400 So we're going to proceed 
with the contrapositive, 123 00:05:19,400 --> 00:05:22,133 so we're going suppose that… 
what's the contrapositive of this? 124 00:05:22,133 --> 00:05:24,333 We're going to take the 
negation of the conclusion. 125 00:05:24,333 --> 00:05:28,566 Well the conclusion is, “a is irrational or b is irrational.” 126 00:05:28,566 --> 00:05:31,000 So what's the negation of 
that? Ask yourself and say, 127 00:05:31,000 --> 00:05:35,300 “Okay well that's a is 
rational and b is rational.” 128 00:05:35,300 --> 00:05:36,733 That's a lot easier 
to deal with. 129 00:05:36,733 --> 00:05:39,800 Dealing with a is rational and b is rational 
and proving something, that's a lot easier. 130 00:05:39,800 --> 00:05:42,066 So we're gonna 
start with that. 131 00:05:42,066 --> 00:05:43,300 What does 
that mean? 132 00:05:43,300 --> 00:05:46,166 Well there exists 
integers k, l, m, n 133 00:05:46,166 --> 00:05:51,300 such that a equals k over l, and b 
equals m over n, with l and n not 0. 134 00:05:51,300 --> 00:05:53,633 Then what? Well okay what are we trying to show? 135 00:05:53,633 --> 00:05:55,500 Let’s making sure that we understand 
what we're trying to show. 136 00:05:55,500 --> 00:05:57,933 So we started with the negation of the 
conclusion. We're trying to show 137 00:05:57,933 --> 00:06:00,400 the negation of the 
hypothesis is true. 138 00:06:00,400 --> 00:06:03,300 Well what's the hypothesis? 
The hypothesis is: 139 00:06:03,300 --> 00:06:05,600 “a b is irrational.” Don't... 140 00:06:05,600 --> 00:06:07,966 worry about the a and b, 
inside the real numbers. That's 141 00:06:07,966 --> 00:06:09,600 sort of… that's above the 
statement, right? This is like a 142 00:06:09,600 --> 00:06:11,866 “for all a and b inside 
the real numbers, 143 00:06:11,866 --> 00:06:15,033 this implies this.” That's the 
way you should read this. 144 00:06:15,033 --> 00:06:19,300 So here we want to show the negation 
the hypothesis which is: “a b is rational”. 145 00:06:19,300 --> 00:06:21,800 Well what does it mean for a b to 
be rational? Well I can write it as 146 00:06:21,800 --> 00:06:24,266 some integer divided 
by some other integer, 147 00:06:24,266 --> 00:06:26,366 and the bottom 
integer is not 0. 148 00:06:26,366 --> 00:06:30,133 Well if l and n weren’t 0, the 
product is definitely not 0. 149 00:06:30,133 --> 00:06:33,000 All of these are products of 
integers, so they’re still an integer, 150 00:06:33,000 --> 00:06:37,366 therefore a times b is going to be 
an integer over a non-zero integer, 151 00:06:37,366 --> 00:06:39,600 that’s a rational, and we’re done. 152 00:06:39,600 --> 00:06:41,633 So there's an example of 
the use of contrapositive. 153 00:06:41,633 --> 00:06:44,500 When the conclusion contains the negation of 154 00:06:44,500 --> 00:06:46,900 an existence sort of thing. 155 00:06:46,900 --> 00:06:50,833 Again, just rules of thumb, when you use it, try it, and see if it works. 156 00:06:50,833 --> 00:06:54,700 If it works then you should use it. If it 
doesn't work, then try something else. 157 00:06:54,700 --> 00:06:58,466 Again, our toolbox of techniques 
are getting bigger, okay? 158 00:06:58,466 --> 00:07:00,500 So when you see 
a new problem, 159 00:07:00,500 --> 00:07:03,900 it's not going to be as obvious 
anymore what tool to use. 160 00:07:03,900 --> 00:07:07,433 In real life, you know it's very easy 
to know what screwdriver to use, 161 00:07:07,433 --> 00:07:10,866 or maybe not so obvious what wrench to 
use, maybe that's a better analogy here. 162 00:07:10,866 --> 00:07:13,300 Figuring out the size of the 
wrench to use can be tricky. 163 00:07:13,300 --> 00:07:15,600 You have to actually measure 
and try some things. 164 00:07:15,600 --> 00:07:16,366 165 00:07:16,366 --> 00:07:17,833 But that's the 
idea now, okay? 166 00:07:17,833 --> 00:07:21,999 So we have a lot of tools. Now we have a wrench with lots of different, you know, socket pieces.