1 00:00:00,600 --> 00:00:05,233 In this video, I'd like to talk about 
a common misconception that 2 00:00:05,233 --> 00:00:08,500 people have in their first proofs course, 3 00:00:08,500 --> 00:00:12,066 and in this example, we are going 
to do an inequalities example, 4 00:00:12,066 --> 00:00:12,766 5 00:00:12,766 --> 00:00:14,766 and the question states, 6 00:00:14,766 --> 00:00:16,833 “Show that for all 7 00:00:16,833 --> 00:00:17,333 8 00:00:17,333 --> 00:00:19,066 real numbers x and y 9 00:00:19,066 --> 00:00:23,666 that 9x to the power of 4, plus 4y squared 10 00:00:23,666 --> 00:00:29,033 plus x squared is greater than or equal to 12x squared times y. 11 00:00:29,033 --> 00:00:31,033 12 00:00:31,033 --> 00:00:34,900 And I'm going to present a proof of this fact 13 00:00:34,900 --> 00:00:38,933 and the proof will go as follows. For all x and y in the real numbers, we have 14 00:00:38,933 --> 00:00:39,700 15 00:00:39,700 --> 00:00:44,000 9x to the power of 4, plus 4y squared, plus x squared is greater than or equal to 12, 16 00:00:44,000 --> 00:00:44,700 17 00:00:44,700 --> 00:00:49,233 and then in the next line we've brought the 12x squared y to the other side. 18 00:00:49,233 --> 00:00:50,533 19 00:00:50,533 --> 00:00:53,700 Then I've moved it between these two… 20 00:00:53,700 --> 00:00:57,200 well these two terms, 9x to 
the power 4 and 4y squared. 21 00:00:57,200 --> 00:01:00,566 Then I've taken the first three terms 
and I factor them to 3x squared 22 00:01:00,566 --> 00:01:03,000 minus 2y all squared. 23 00:01:03,000 --> 00:01:05,533 Keep in mind that this is a trinomial, so this is 24 00:01:05,533 --> 00:01:08,133 3x squared all squared, 25 00:01:08,133 --> 00:01:10,600 this is 2y all squared, 26 00:01:10,600 --> 00:01:11,266 27 00:01:11,266 --> 00:01:13,200 and this is the cross term that you get. 28 00:01:13,200 --> 00:01:14,033 29 00:01:14,033 --> 00:01:18,366 And this last line is true. So here's 
the proof that I'm presenting to you. 30 00:01:18,366 --> 00:01:21,700 What I'd like you to do now 
is pause the video and 31 00:01:21,700 --> 00:01:24,933 just ask yourself is this proof 
correct or is this proof incorrect? 32 00:01:24,933 --> 00:01:25,500 33 00:01:25,500 --> 00:01:29,200 And hopefully you've done that, 
and you've critiqued this proof, 34 00:01:29,200 --> 00:01:32,366 and hopefully you've come to the 
conclusion this proof is not correct, and 35 00:01:32,366 --> 00:01:36,200 the fundamental flaw with this 
proof is that it's written backwards. 36 00:01:36,200 --> 00:01:38,400 Mathematics is read top to bottom, 37 00:01:38,400 --> 00:01:41,866 so the way we read this 
is we start with line 1, 38 00:01:41,866 --> 00:01:44,800 and we say that line 1 implies that line 2 is true, 39 00:01:44,800 --> 00:01:48,600 which implies that line 3 is true 
which implies that line 4 is true. 40 00:01:48,600 --> 00:01:50,866 However line 1 is a line 41 00:01:50,866 --> 00:01:54,666 that we don't know if it's true or false, that's 
what the question is asking us to prove. 42 00:01:54,666 --> 00:01:57,066 So proving a bunch of implications 43 00:01:57,066 --> 00:01:59,333 without knowing that the first term is true 44 00:01:59,333 --> 00:02:01,466 is not logically valid as an argument. 45 00:02:01,466 --> 00:02:02,300 46 00:02:02,300 --> 00:02:05,900 There are two ways to correct this proof… well 
there are lots of ways to correct this proof, 47 00:02:05,900 --> 00:02:08,000 but I will show you two different ways 48 00:02:08,000 --> 00:02:10,100 right now. 49 00:02:10,100 --> 00:02:13,500 By the way before I do, again 
a good exercise is to try to 50 00:02:13,500 --> 00:02:17,266 correct this proof for yourself first, and then continue watching the video. 51 00:02:17,266 --> 00:02:20,000 52 00:02:20,000 --> 00:02:21,300 53 00:02:21,300 --> 00:02:24,366 Here's a correct solution 
to this problem. So, 54 00:02:24,366 --> 00:02:28,300 for all real numbers x and y we have 55 00:02:28,300 --> 00:02:32,766 that 3x squared minus 2y all squared, plus 
x squared is greater than or equal to 0, 56 00:02:32,766 --> 00:02:36,666 and that is a true statement 
because this is a non… 57 00:02:36,666 --> 00:02:40,433 this is a real number and this is a real number. When we square real numbers, 58 00:02:40,433 --> 00:02:42,166 we get non-negative numbers, 59 00:02:42,166 --> 00:02:45,700 and when I add two non-negative 
numbers, my result is still non-negative. 60 00:02:45,700 --> 00:02:46,900 61 00:02:46,900 --> 00:02:50,866 Further, this implies that if I just 
expand this out and then rearrange, 62 00:02:50,866 --> 00:02:54,600 I get the result that I actually wanted 
at the beginning. So that works. 63 00:02:54,600 --> 00:02:56,900 64 00:02:56,900 --> 00:02:59,800 For another proof, what you 
can do is you can start with 65 00:02:59,800 --> 00:03:02,500 the inequality but instead of 
having an implication here, 66 00:03:02,500 --> 00:03:05,833 which is implicit in the original 
proof that I gave you, 67 00:03:05,833 --> 00:03:09,933 we can make it a bidirectional implication. 
In other words, an “if and only if” statement. 68 00:03:09,933 --> 00:03:12,966 So this is true if and only if this is true. 69 00:03:12,966 --> 00:03:16,500 Why is that valid? Again, 
we can move a term to the 70 00:03:16,500 --> 00:03:20,000 other side and just switch the sign, 
and we can bring the term back over, 71 00:03:20,000 --> 00:03:23,266 and once again just switch the sign, that's okay. 72 00:03:23,266 --> 00:03:23,700 73 00:03:23,700 --> 00:03:26,066 So these two things imply each 
other. Make sure though that 74 00:03:26,066 --> 00:03:29,466 both directions work. So here we're moving terms around, that's okay, 75 00:03:29,466 --> 00:03:33,266 you have to be careful though when you're 
doing things like multiplication and division. 76 00:03:33,266 --> 00:03:36,066 One direction might be easy, the other direction might not be possible. 77 00:03:36,066 --> 00:03:39,366 For example, we could multiply 
by 0 but we can't divide by 0. 78 00:03:39,366 --> 00:03:40,600 79 00:03:40,600 --> 00:03:44,600 Once you have this thing, this is 
true if and only if this fact is true, 80 00:03:44,600 --> 00:03:45,433 81 00:03:45,433 --> 00:03:48,966 and again how do we do this? Well we took 
these three terms and we factored them, 82 00:03:48,966 --> 00:03:50,766 or we took this term and expanded it. 83 00:03:50,766 --> 00:03:54,633 Both of those operations are reversible, 
so that's okay in this situation as well. 84 00:03:54,633 --> 00:03:58,066 The last statement is true, all of 
these are “if and only if” statements, 85 00:03:58,066 --> 00:04:01,500 so therefore this one is true if and only 
if this is true if and only if this is true. 86 00:04:01,500 --> 00:04:02,233 87 00:04:02,233 --> 00:04:04,800 And so you're done, this 
is a perfectly valid proof. 88 00:04:04,800 --> 00:04:08,500 So again, the key idea here is that we've 
gone to an “if and only if” statement 89 00:04:08,500 --> 00:04:10,533 instead of just an implication. 90 00:04:10,533 --> 00:04:11,666 91 00:04:11,666 --> 00:04:13,600 Okay... 92 00:04:13,600 --> 00:04:15,533 so hopefully I gave you a little bit 93 00:04:15,533 --> 00:04:18,533 of insight into this 94 00:04:18,533 --> 00:04:21,466 problem. Thank you very 
much for your attention.