>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
This is part 2
of Venn diagrams.
We'll just continue doing some
more complicated problems
with Venn diagrams.
All right,
so let's take this example.
How about A union B minus C?
All right,
so we've got a Venn diagram
with three sets.
[ Pause ]
And I want to do A union B
but take any part of C out.
So first of all,
we have to think
about where A union B is.
All right, so that's all of A
and all of B.
So hopefully you could see it
would be everything in here.
Are we okay with that?
I didn't shade it yet.
But I don't want
to include any part of C.
So right in here,
this little place
where I got some dots,
that's part of C.
So that can't be part
of my shading.
So my shading then is this
part of A union B
but I can't include this part
that has C, okay?
Now, another way,
if you have trouble doing
that, seeing it
that way is simply we'd just
go ahead and shade all
of A union B, okay?
So we shade all
of A union B one way
and then we're going
to shade C a different way.
So I'm going to shade all
of C a different way
and my final answer has
to only look like this, okay?
So where's the only part
that only looks
like this 'cause I only want
the part of A union B
that doesn't have anything
else in it, right,
where I didn't take some
of it off.
So, I could see
that this part here, right,
this can't be part of it
and in here.
This can't be part of it
so you can get rid
of that stuff in here
and there we go.
You've got A union B,
take away C. So it's one way
of figuring it out.
Everyone has
different strategies.
All right,
here's our next example.
A intersect B and union
that was C. All right,
so here's how I'm going
to figure this one out.
I'm going to just do A
intersect B that direction,
all right,
and at the same time I'm going
to do C that direction.
So I'm going
to have everything
that direction.
So where is A intersect B?
So I have to concentrate on A
and B here
and where do they intersect?
It's right here in the middle.
Okay, so I know that's part
of my shading
because I'm unioning
that with something else.
So all of that gets to stay
in the answer and now,
here's all of C,
just also shade all of C.
And there we go.
So that's what A union B,
I mean I'm sorry,
A intersect B union C
looks like.
Here's our next example,
it's the difference, right?
So we have A minus B intersect
C. So I know my final answer
is going to have no more
than A, right?
So now my final answer is
going to be maybe something
in A, but certainly nothing
outside of A. Take away B
intersect C. So what I'm going
to do is, say, well,
if A was shaded this way,
let's start off with that,
A shaded this way.
I want to take away anything
that's in B intersect C. Now,
what part of that is in B
and intersect C?
Well, right here is your B
intersect C, see that?
So I have to take away this
little part right in here.
Okay, I got to take that off,
that's the idea.
So, I'm going
to take that off.
Okay, get rid of that,
but keep in mind,
this part was still here,
okay?
So, I only have some
of A shaded
but I took away this part
here, this part of A
where that was part
of B intersect C. Tricky.
All right,
let's try another one.
So let's do this one,
A intersect B,
union B intersect C. All
right, so,
I want to shade A intersect B
and I also want
to shade B intersect C
and that will be my
final answer.
So where is A intersect B?
So here's A, here's B,
where's their intersection?
Right here in the middle.
All right, we also want
to do B intersect C.
So now we concentrate
on the set C and the set B
and their intersection is
right here so we're going
to add just a little bit here.
And there we've got A
intersect B,
union B intersect C. This one
is a little tricky,
A minus B complement.
So here's our Venn
diagram again.
[ Pause ]
We're going
to just totally ignore the C
part, right?
It has something to do
with this.
So we're going
to have A minus B complement,
that's hard to figure out all
in one step.
So what I'm going
to do is I'm going to go ahead
and shade all of A, okay?
Actually, I'm going
to shade all of B first.
I'm sorry.
I'm going to shade
B complement.
So where is B complement?
It's everything outside
of B, right?
[ Pause ]
All of this stuff,
shading everything
that B. My answer should be A
minus anything I've shaded.
All right,
so now I'll concentrate
on where A is.
You see A right here?
This is A and my answer,
I can't have anything
where I've got this blue part.
So what is left?
What's left is just this right
here, isn't it?
What's an easier
of writing that,
this part where I just
shade this?
So here's my answer, okay?
So the purple is my answer.
[ Pause ]
Do you recognize
that as something else?
That's actually A intersect B.
So this Venn diagram shows you
that A take away B complement
and A intersect B is exactly
the same thing,
kind of interesting.
In any case,
this is the shading
right here.
In our problems before I went
to Venn diagrams,
we found out that A union B
complement was the same thing
as A complement intersection--
intersect B complement.
We could show
that with Venn diagrams.
If I want to do A union B
complement,
think about where the union
of A and B is, right here,
and then I want everything
outside of that.
So I'm going
to shade everything outside
of A union B. In fact,
let me get rid of C completely
because we could prove my
point a little bit easier
by just concentrating on A
and B. So we have AB,
we're going
to shade everything outside
of A union B. Okay.
Now, see if this ends
up looking the same.
I want to do the intersection
of A complement
and B complement.
So I'm going
to have an intersection here,
right?
So if were going
to shade everything outside
of A, I would have all
of this crosshatch, right?
Everything outside of A,
I'm going to go one direction
and everything outside of B,
I'm going to do
with other direction.
[ Pause ]
Okay.
What did I do here?
[ Pause ]
What did I do
for everything outside of A?
I forgot to keep shading,
here we go.
Outside of A,
I forgot to keep going.
Here we go, sorry,
and then something
like done here.
Okay, so I shaded everything
outside of A, one direction,
everything outside of B
with a different direction,
and then where they meet
in the middle,
that's the intersection.
So you see
where all the
crosshatching is?
It's outside of where A
and B are, right?
It's outside, the outside
of that, so these actually are
equal to each other.
So you can actually prove this
using a Venn diagram.
Okay, Venn diagrams can get
super complicated,
but I think we've done several
examples that should help you.
You just have
to be pretty careful
with your shading
and your strategies.
[ Silence ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.