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>> Julie Harland: Hi,
this is Julie Harland
and I'm Your Math Gal.
Please visit any web site
at YourMathGal.com
where you can search for any
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by topic.
This is an introduction
to Venn diagrams,
so you already have
to know a little bit
about sets.
What we're going
to do is draw a box,
a rectangle, if you will,
to represent the
universal set.
Okay, remember
that like contains everything
in this box.
And let's say there's some
other sets
like there's the set A. That's
a subset of U.
So this is the set A.
So this shows you
that A is a subset
of the universe.
Now, let's say I wanted
to know what A comp --
so I want to know where I was,
you would just shade A. Now,
you could, you know,
use a shading like this,
a different color, okay.
And so this represents
the picture.
The shading represents what
A is.
Another way people do this,
because you don't always
happen to have colors.
This is universal,
call this A --
is you just shade it this way,
you just put some lines
through it
to show that's what A is.
How about if I wanted
to do A complement?
That means I want everything,
except A. So that means I have
to shade everything outside
of A that's in the universe.
So this is a picture
of what A complement
looks like.
So this is A complement.
So we have A
and we have A complement.
All right, how about you do --
look at the picture
and you call this B,
call this the universe,
and I want you
to shade B complement.
What would you do?
Try it. You would just shade
outside here,
everything here is
B complement.
Okay we're going to go
on to intersection.
So let's say we have two sets.
And I'm showing there might be
some overlap.
For instance,
maybe A represents all the
people who have cats
in their house
and B represents all the
people who have dogs
in their house.
Right? Well,
some people have both cats
and dogs, right?
And then some people don't
have any animals
or don't have any dogs
and cats at least.
So that might be an example,
you know, why there's
some overlap.
And I want
to do A intersect B.
That means where's the place
they have in common?
So it's this part
in the middle
because this part right here,
okay, is part of A
and is also part
of B. Now another way
to get the intersection is
to shade one of the --
wait, let me move this a
little bit here.
If you shade one of the sides
of the intersection
crosshatched one way
and the other a different way,
you're going to look and see
where they both cross.
That will be the intersection.
So if I do A this way,
all of A. Now you have
to ignore everything,
but just all of A,
that whole circle gets shaded
this way, and then all
of B gets shaded this way.
The crosshatch,
the intersection is
where they both get shaded.
So this yellow part is
the intersection.
Now, so let's say you were
taking a test
and somebody said shade the
intersection,
you wouldn't want to show
that all of that here.
You would say the answer,
that's sort
of like your scratch work,
you would just shade
that part.
So this represents A intersect
B. Now, what
if I wanted A union B?
I would actually --
I could do the same shading,
A crosshatched one way
and B the other way,
and it would look the same,
except your answer would be
where there's any shading
at all, so it would be all
of A and B. That's why you
want to show your final answer
because I can't tell what you
think the answer is unless you
happen to have
like this highlighter yellow,
for instance.
So let's do that.
Let's do A union B. How
about we do this, so we have A
and B. And since it's the
union, I'm just going
to shade all of A
and then I'm also going
to shade all of B and then
where there's any shading
at all, that's the answer.
So in this case, every place
where you have a shading is
the answer.
Now, if you would have
crosshatched the other,
it might be confusing
to people.
They might think
that you think this little
part in the middle is
the answer.
So that's why I did just the
same kind of shading
both ways.
And that's A union B. Now
let's try doing a problem
where there are three sets, A,
B and C. So let's say I want
to do A intersect C. All
right, so one way
of doing this is I could say A
one direction,
C the other direction,
and then I'm going to see
where there's a crosshatching,
right, for both of them.
So if I shade A with all
of this, just ignore B
and C. Just shade all
of A one direction,
and then I'm going
to shade C the
other direction.
And now I'm going
to see whether there's
crosshatching
for both of them.
And that would be right here
in the middle,
just this little part here,
okay?
So you got to be careful,
again, this is just sort
of like how I figure
out the answer to do that.
And some of you, I'm sure,
wouldn't even have
to do all of that.
So now, A intersect C is just
this part right here, okay?
Just want to make sure you get
that, it's only this.
And you could just fill it all
in, you know,
how ever you want to do that.
There's A intersect C. Okay,
so what I want you
to do is A union B
when you've got three circles,
A, B and C. All right,
so do that on your own.
So this means you want all
of B and all of A.
So I'm just going to get all
of A. See how I've got all
of A?
Now I'm just going
to just concentrate
on the circle B
and I'm just going
to shade all of B as well.
And all together,
that will give me the complete
shading for A union B. Now
what if the problem had been A
union B complement?
All right,
so what would A union B
complement look like?
Think about that.
You see where A union B is,
right?
So the complement
of this would be everything
outside of A union B.
So what would that look like?
So you have A,
B and C. I want everything,
except the shading I see here,
so I'm going
to shade everything outside.
So it's going to get part
of C, isn't it?
And no part of A
or B gets shaded at all.
So look at this.
This is the part right here
that's A union B. It's
everything outside of that.
So this is A union
B complement.
All right,
see if you could figure
out where A intersects C
complement is.
First, of course, you'll have
to know where A intersect C is
so you'll be able
to do the complement.
I forgot, you always want
to put U to show that's the
whole universe there.
Okay, so where would A
intersect C be?
Well, it would be
that part inside here.
Right, see how that's the part
that has both A and C in it?
So what you're going
to do is shade everything
outside of it,
everything except
that little football
shape there.
What about A takeaway B?
Which also, remember,
it can be written
as A minus B.
So it's a difference of A
and B. So this means we want
all of A, except the part
that's in B. Can you figure
that one out,
what that would look like?
All of A except what's
in B. Well, here's all of A,
right, but this part here
in the middle is what's also
in B. So that is going
to be shading only this part
of A. So that's A takeaway B.
Now what if C had been
in here?
It still looks exactly
the same.
You ignore C, right?
It has nothing to do with it.
You concentrate
where A takeaway B is.
Now what if I wanted A
takeaway B complement,
you would shade everything
except what I have here
in green and so on.
So I'll do some more
complicated problems like that
on the next video.
[ Silence ]
Please visit my web site
at YourMathGal.com
where you can view all
of my videos
which are organized by topic.