1 00:00:00,000 --> 00:00:04,000 Hello everyone. 
In this screencast, 2 00:00:04,000 --> 00:00:06,266 we're going to talk 
about uniqueness. 3 00:00:06,266 --> 00:00:08,200 So I have two 
examples here 4 00:00:08,200 --> 00:00:12,000 that are usually found inside 
group theory and ring theory, 5 00:00:12,000 --> 00:00:16,433 but I'm gonna use them just to 
emphasize the purpose of uniqueness, 6 00:00:16,433 --> 00:00:19,466 and I'll do this over the real 
numbers so that we have an idea. 7 00:00:19,466 --> 00:00:23,266 So, the first of the 
questions states here, 8 00:00:23,266 --> 00:00:25,333 "Show that there's a unique real number y such that 9 00:00:25,333 --> 00:00:28,300 x y equals x for all real numbers x. 10 00:00:28,300 --> 00:00:31,800 So this example, 
you can 11 00:00:31,800 --> 00:00:33,933 kind of intuitively tell yourself, "Okay 12 00:00:33,933 --> 00:00:37,400 this is trying to show you that 
there's a unique '1' element" right? 13 00:00:37,400 --> 00:00:41,400 There's only…I mean y here's supposed to take the value 1 basically. 14 00:00:41,400 --> 00:00:44,633 The second question: "Show there 
is a unique real number y such that 15 00:00:44,633 --> 00:00:47,533 x plus y equals x for 
all real numbers x." 16 00:00:47,533 --> 00:00:51,833 This is sort of saying there's a 
unique '0' element in the real numbers. 17 00:00:51,833 --> 00:00:55,133 That's intuitively what you should read these as but the proof 18 00:00:55,133 --> 00:00:57,133 of these things is gonna 
be a little different. So, 19 00:00:57,133 --> 00:00:59,500 how are we proving 
these things? 20 00:00:59,500 --> 00:01:02,066 It doesn't say anything 
about existence so... 21 00:01:02,066 --> 00:01:04,000 I mean, we know 
that they exist, 22 00:01:04,000 --> 00:01:05,733 but it says about 
uniqueness, right… 23 00:01:05,733 --> 00:01:08,466 well okay, so I guess existence 
is sort of implied but again, 24 00:01:08,466 --> 00:01:12,000 we already talked about 
1 and 0 being there. 25 00:01:12,000 --> 00:01:16,566 Notice that 
y equals 1 26 00:01:16,566 --> 00:01:16,600 27 00:01:16,600 --> 00:01:20,366 is a possible 
value for y 28 00:01:20,366 --> 00:01:22,566 So let's just write that sentence in just so that we can talk about 29 00:01:22,566 --> 00:01:25,500 the more important part of this pencast, which is uniqueness. 30 00:01:25,500 --> 00:01:30,733 Now, suppose there 
are two values 31 00:01:30,733 --> 00:01:34,833 say, y and z, 32 00:01:34,833 --> 00:01:36,433 such that 33 00:01:36,433 --> 00:01:38,066 34 00:01:38,066 --> 00:01:41,766 x y equals x and 
x z equals x 35 00:01:41,766 --> 00:01:45,900 for all values, 36 00:01:45,900 --> 00:01:49,433 let's say 
real, x. 37 00:01:49,433 --> 00:01:52,233 "Now suppose there are two 
values, say y and z, such that 38 00:01:52,233 --> 00:01:55,133 x y equals x and x z equals x for all real x." 39 00:01:55,133 --> 00:01:57,033 Well now they're both 
equal to x, so I 40 00:01:57,033 --> 00:01:59,533 should be able to subtract 
the two equations so... 41 00:01:59,533 --> 00:02:02,033 42 00:02:02,033 --> 00:02:06,433 yields x y minus 
x z equals 0. 43 00:02:06,433 --> 00:02:07,533 44 00:02:07,533 --> 00:02:08,900 45 00:02:08,900 --> 00:02:14,200 Factoring gives x, y 
minus z equals 0, 46 00:02:14,200 --> 00:02:15,466 47 00:02:15,466 --> 00:02:19,000 which holds 
for all real x. 48 00:02:19,000 --> 00:02:20,933 So let's look 
what this says so, 49 00:02:20,933 --> 00:02:25,233 "Subtracting yields x y 
minus x z equals 0…", 50 00:02:25,233 --> 00:02:28,000 I guess I should use 
correct punctuation 51 00:02:28,000 --> 00:02:30,900 so let's put a period there 
and put a capital F here, 52 00:02:30,900 --> 00:02:34,900 "…factoring gives x 
times y minus z 53 00:02:34,900 --> 00:02:38,266 equals 0 which 
holds for all real x. 54 00:02:38,266 --> 00:02:41,266 So if we take x to be non-zero, then, what does this mean? 55 00:02:41,266 --> 00:02:43,333 Well it must mean that 
y minus z must be 0. 56 00:02:43,333 --> 00:02:46,166 Since this has to hold for 
every single value of x, 57 00:02:46,166 --> 00:02:49,300 you could plug in, let's say, 
x equals 1 or x equals 2 58 00:02:49,300 --> 00:02:51,100 it has to work for everything, right? 59 00:02:51,100 --> 00:02:54,900 So we really do know that 
y minus z must be 0 here. 60 00:02:54,900 --> 00:02:56,333 61 00:02:56,333 --> 00:02:58,900 So how can 
I write that? 62 00:02:58,900 --> 00:02:58,933 63 00:02:58,933 --> 00:02:59,000 64 00:02:59,000 --> 00:03:02,166 So in particular, taking x to be non-zero, we see that 65 00:03:02,166 --> 00:03:04,966 y minus z must equal 
0, and hence y equals z. 66 00:03:04,966 --> 00:03:07,166 Again, this is valid because 
we have to have this hold for 67 00:03:07,166 --> 00:03:12,033 every single real value of x so it has to 
hold for, in particular, non-zero values of x. 68 00:03:12,033 --> 00:03:14,800 Let's look at 
the second one… 69 00:03:14,800 --> 00:03:17,633 maybe I'll put a couple of 70 00:03:17,633 --> 00:03:21,400 spaces in there just so that we can separate the two… 71 00:03:21,400 --> 00:03:22,666 72 00:03:22,666 --> 00:03:24,866 okay so the second one you're gonna 
just go the same way, right, so 73 00:03:24,866 --> 00:03:26,700 what's the key idea here with uniqueness? 74 00:03:26,700 --> 00:03:28,433 Now suppose there are two values, 75 00:03:28,433 --> 00:03:30,533 if you want to prove something's 
unique, suppose two exist, 76 00:03:30,533 --> 00:03:32,566 and then show that 
that can't happen, 77 00:03:32,566 --> 00:03:33,966 78 00:03:33,966 --> 00:03:37,800 or show that if you have two of the 
values then they must be equal. 79 00:03:37,800 --> 00:03:39,266 So that's what 
we did there. 80 00:03:39,266 --> 00:03:41,900 I'm gonna probably write 
this one the same way 81 00:03:41,900 --> 00:03:44,566 but again, you can also derive a contradiction 82 00:03:44,566 --> 00:03:48,033 assuming that y and z aren't 
equal and using that somehow. 83 00:03:48,033 --> 00:03:50,300 84 00:03:50,300 --> 00:03:52,833 So again, notice first in the second example, 85 00:03:52,833 --> 00:03:54,700 "Show that there is a unique 
real number y such that 86 00:03:54,700 --> 00:03:56,566 x plus y equals x for 
all real numbers x." 87 00:03:56,566 --> 00:03:59,633 Notice that 
y equals 0 88 00:03:59,633 --> 00:04:00,566 Notice that 
y equals 0 89 00:04:00,566 --> 00:04:02,033 90 00:04:02,033 --> 00:04:04,566 is a possible 
solution. 91 00:04:04,566 --> 00:04:09,800 Now suppose that 92 00:04:09,800 --> 00:04:11,800 there exists a 93 00:04:11,800 --> 00:04:14,800 94 00:04:14,800 --> 00:04:17,166 y and z in the 
real numbers 95 00:04:17,166 --> 00:04:18,866 96 00:04:18,866 --> 00:04:20,866 such that 97 00:04:20,866 --> 00:04:23,700 98 00:04:23,700 --> 00:04:28,566 x plus y equals x 
and x plus z equals x. 99 00:04:28,566 --> 00:04:31,000 Subtracting these yields 100 00:04:31,000 --> 00:04:36,433 y minus z equals 0 
and hence, y equals z. 101 00:04:36,433 --> 00:04:38,000 102 00:04:38,000 --> 00:04:40,833 So the second 
example is a lot easier, 103 00:04:40,833 --> 00:04:43,133 104 00:04:43,133 --> 00:04:47,100 but that's okay. Again, it's the point of these questions that's important, right? 105 00:04:47,100 --> 00:04:49,000 So in the 
second example, 106 00:04:49,000 --> 00:04:52,666 again, y equals 0 is a possible 
solution. Now suppose that we have 107 00:04:52,666 --> 00:04:55,800 two values, you 
subtract them, 108 00:04:55,800 --> 00:04:59,833 and then you get 1 minus z equals 0, and hence y must equal z. 109 00:04:59,833 --> 00:05:03,666 Again, these are just the types of examples for uniqueness. 110 00:05:03,666 --> 00:05:03,700 111 00:05:03,700 --> 00:05:06,333 Like I said at the 
beginning of the screencast, 112 00:05:06,333 --> 00:05:08,433 these will make 
more sense in 113 00:05:08,433 --> 00:05:12,433 group theory and ring theory, if you ever take these classes, 114 00:05:12,433 --> 00:05:16,033 but because they hear that proofs 
aren't quite as easy as they are here, 115 00:05:16,033 --> 00:05:20,200 but nonetheless, this gives you an idea of how to show that something is unique. 116 00:05:20,200 --> 00:05:21,866 Assume that there 
are two of them, 117 00:05:21,866 --> 00:05:24,666 and then show either that 
they must be equal, or 118 00:05:24,666 --> 00:05:28,433 you assume that they're not 
equal and derive a contradiction. 119 00:05:28,433 --> 00:05:33,066 Hopefully that gives you an idea, 
thank you very much for listening. 120 00:05:33,066 --> 00:05:33,099