In this video, I'd like to talk about a common misconception that
people have in their first proofs course,
and in this example, we are going to do an inequalities example,
and the question states,
“Show that for all
real numbers x and y
that 9x to the power of 4, plus 4y squared
plus x squared is greater than or equal to 12x squared times y.
And I'm going to present a proof of this fact
and the proof will go as follows. For all x and y in the real numbers, we have
9x to the power of 4, plus 4y squared, plus x squared is greater than or equal to 12,
and then in the next line we've brought the 12x squared y to the other side.
Then I've moved it between these two…
well these two terms, 9x to the power 4 and 4y squared.
Then I've taken the first three terms and I factor them to 3x squared
minus 2y all squared.
Keep in mind that this is a trinomial, so this is
3x squared all squared,
this is 2y all squared,
and this is the cross term that you get.
And this last line is true. So here's the proof that I'm presenting to you.
What I'd like you to do now is pause the video and
just ask yourself is this proof correct or is this proof incorrect?
And hopefully you've done that, and you've critiqued this proof,
and hopefully you've come to the conclusion this proof is not correct, and
the fundamental flaw with this proof is that it's written backwards.
Mathematics is read top to bottom,
so the way we read this is we start with line 1,
and we say that line 1 implies that line 2 is true,
which implies that line 3 is true which implies that line 4 is true.
However line 1 is a line
that we don't know if it's true or false, that's what the question is asking us to prove.
So proving a bunch of implications
without knowing that the first term is true
is not logically valid as an argument.
There are two ways to correct this proof… well there are lots of ways to correct this proof,
but I will show you two different ways
right now.
By the way before I do, again a good exercise is to try to
correct this proof for yourself first, and then continue watching the video.
Here's a correct solution to this problem. So,
for all real numbers x and y we have
that 3x squared minus 2y all squared, plus x squared is greater than or equal to 0,
and that is a true statement because this is a non…
this is a real number and this is a real number. When we square real numbers,
we get non-negative numbers,
and when I add two non-negative numbers, my result is still non-negative.
Further, this implies that if I just expand this out and then rearrange,
I get the result that I actually wanted at the beginning. So that works.
For another proof, what you can do is you can start with
the inequality but instead of having an implication here,
which is implicit in the original proof that I gave you,
we can make it a bidirectional implication. In other words, an “if and only if” statement.
So this is true if and only if this is true.
Why is that valid? Again, we can move a term to the
other side and just switch the sign, and we can bring the term back over,
and once again just switch the sign, that's okay.
So these two things imply each other. Make sure though that
both directions work. So here we're moving terms around, that's okay,
you have to be careful though when you're doing things like multiplication and division.
One direction might be easy, the other direction might not be possible.
For example, we could multiply by 0 but we can't divide by 0.
Once you have this thing, this is true if and only if this fact is true,
and again how do we do this? Well we took these three terms and we factored them,
or we took this term and expanded it.
Both of those operations are reversible, so that's okay in this situation as well.
The last statement is true, all of these are “if and only if” statements,
so therefore this one is true if and only if this is true if and only if this is true.
And so you're done, this is a perfectly valid proof.
So again, the key idea here is that we've gone to an “if and only if” statement
instead of just an implication.
Okay...
so hopefully I gave you a little bit
of insight into this
problem. Thank you very much for your attention.