Hello everyone. In this screencast,
we're going to talk about uniqueness.
So I have two examples here
that are usually found inside group theory and ring theory,
but I'm gonna use them just to emphasize the purpose of uniqueness,
and I'll do this over the real numbers so that we have an idea.
So, the first of the questions states here,
"Show that there's a unique real number y such that
x y equals x for all real numbers x.
So this example, you can
kind of intuitively tell yourself, "Okay
this is trying to show you that there's a unique '1' element" right?
There's only…I mean y here's supposed to take the value 1 basically.
The second question: "Show there is a unique real number y such that
x plus y equals x for all real numbers x."
This is sort of saying there's a unique '0' element in the real numbers.
That's intuitively what you should read these as but the proof
of these things is gonna be a little different. So,
how are we proving these things?
It doesn't say anything about existence so...
I mean, we know that they exist,
but it says about uniqueness, right…
well okay, so I guess existence is sort of implied but again,
we already talked about 1 and 0 being there.
Notice that y equals 1
is a possible value for y
So let's just write that sentence in just so that we can talk about
the more important part of this pencast, which is uniqueness.
Now, suppose there are two values
say, y and z,
such that
x y equals x and x z equals x
for all values,
let's say real, x.
"Now suppose there are two values, say y and z, such that
x y equals x and x z equals x for all real x."
Well now they're both equal to x, so I
should be able to subtract the two equations so...
yields x y minus x z equals 0.
Factoring gives x, y minus z equals 0,
which holds for all real x.
So let's look what this says so,
"Subtracting yields x y minus x z equals 0…",
I guess I should use correct punctuation
so let's put a period there and put a capital F here,
"…factoring gives x times y minus z
equals 0 which holds for all real x.
So if we take x to be non-zero, then, what does this mean?
Well it must mean that y minus z must be 0.
Since this has to hold for every single value of x,
you could plug in, let's say, x equals 1 or x equals 2
it has to work for everything, right?
So we really do know that y minus z must be 0 here.
So how can I write that?
So in particular, taking x to be non-zero, we see that
y minus z must equal 0, and hence y equals z.
Again, this is valid because we have to have this hold for
every single real value of x so it has to hold for, in particular, non-zero values of x.
Let's look at the second one…
maybe I'll put a couple of
spaces in there just so that we can separate the two…
okay so the second one you're gonna just go the same way, right, so
what's the key idea here with uniqueness?
Now suppose there are two values,
if you want to prove something's unique, suppose two exist,
and then show that that can't happen,
or show that if you have two of the values then they must be equal.
So that's what we did there.
I'm gonna probably write this one the same way
but again, you can also derive a contradiction
assuming that y and z aren't equal and using that somehow.
So again, notice first in the second example,
"Show that there is a unique real number y such that
x plus y equals x for all real numbers x."
Notice that y equals 0
is a possible solution.
Now suppose that
there exists a
y and z in the real numbers
such that
x plus y equals x and x plus z equals x.
Subtracting these yields
y minus z equals 0 and hence, y equals z.
So the second example is a lot easier,
but that's okay. Again, it's the point of these questions that's important, right?
So in the second example,
again, y equals 0 is a possible solution. Now suppose that we have
two values, you subtract them,
and then you get 1 minus z equals 0, and hence y must equal z.
Again, these are just the types of examples for uniqueness.
Like I said at the beginning of the screencast,
these will make more sense in
group theory and ring theory, if you ever take these classes,
but because they hear that proofs aren't quite as easy as they are here,
but nonetheless, this gives you an idea of how to show that something is unique.
Assume that there are two of them,
and then show either that they must be equal, or
you assume that they're not equal and derive a contradiction.
Hopefully that gives you an idea, thank you very much for listening.