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>> Julie Harland: Hi,
this is Julie Harland
and I'm Your Math Gal.
Please visit my website
at yourmathgal.com
where you can search for any
of my videos organized
by topic.
This is sets part three.
We're going
to do more involved examples
involving intersection, union,
and complement
where it takes more
than one step
and we're using order
of operations
for this universal set
and these two sets at C and F.
And we do these four problems
on this video.
All right.
Let's take this universal set.
So the universe is the numbers
1 through 9.
And we have two subsets I've
created --
one is E containing 1, 3, 5,
and 7; and the other is F
containing 4, 5, and 6.
So how would we do these
two problems?
So the first one is E union
F complement.
Well, this takes more
than one step.
I have to use my order
of operation.
So I first have
to do E union F. All right.
So let's write
that parenthesis
and E union F is a set.
So I need to put it
in braces, right?
And let's list the numbers
in E union F. All right.
So let's look at E and F.
And what are all the numbers?
You put them together.
I see that 5 is listed twice
but all the rest are unique.
So I've got 1, 3,
I also have 4 because that's
in F, right.
I have 5, I have 6, and 7.
So we did the union
in the previous video.
So make sure
that you remember how
to get the union.
And now I'm going
to put a parenthesis
around that.
In other words,
I'm now writing E union F
as this set that it really is,
these numbers, 1, 3, 4,
5, 6, and 7.
And I want the complement
of that.
All right.
Now what's the complement?
It means all the numbers
in the universe except these
numbers, right?
So what does that give me?
Look at the universal set
what's not in here,
all right --
a 2, an 8, and a 9.
So that is E union
F complement.
Now remember
when you put these two
together, right,
if you have a set
and its complement,
then all together they should
make up the universal set,
right?
And they should be different
numbers of course.
All right,
let's do the second part.
How about E complement union
F complement?
This is not the same problem.
There's no parenthesis.
So first I have
to do E complement,
and I have to do F complement,
and then I need
to take their union.
So let's do that.
First we've got
to take E complement.
So we look up here
at E. We want everything
in the universal set except 1,
3, 5, and 7.
So that's this set --
2, 4, 6, 8, and 9.
And I'm writing
that union symbol.
And now I need
to do F complement --
that's everything
in the universe except 4,
5, and 6.
So that's 1, 2, 3, 7, 8, 9.
Remember you don't have
to write the numbers in order,
but sometimes it's kind
of confusing
if you don't write them
in order.
So that's why I'm writing them
in order from smallest
to largest.
But the order does not matter.
Now, I need
to take their union.
So if it's in one
of these sets it's
in the union, right?
And if I want to write them
in order from 1
to 9 I'm just going to start
with the number 1 and see
if it's in at least one
of the sets.
So is 1 in the union?
Yes, because it's
in F complement over here.
Is 2 in? Yes.
How about 3?
Yes, there's a 3 somewhere.
How about 4?
Yes. Is there a 5?
No. So 5's not in the union.
How about 6?
Yes, there's a 6.
7 is in there.
There's an 8 and there's a 9.
Now, someone else might have
done it this way --
they may just listed the first
set which is 2, 4, 6, 8,
and 9, and then listed the
numbers in the second set
but just don't
[inaudible] list.
So 1 not in there yet,
2's already there,
3's not in there,
7 not in there yet,
8 it's already in there,
9's already in there.
Okay. So those should be the
same number, the elements,
the same exact elements.
Let's say I've got 1, 2, 3, 4,
5, 6, 7, 8 numbers here.
1, 2, 3, 4, 5, 6, 7 --
yeah, the same eight numbers.
So you how figure
that out doesn't matter.
So what I want you
to notice here, though,
is that these are definitely
not the same.
E union F complement is not
the same.
So it's not equal
to E complement union
F complement.
All right, so now I want you
to try a different example.
I want you to start off
with the same universal set
and E and F.
And what we're going
to do is E intersect F
in parenthesis complement
and then E complement
intersect F complement.
All right.
So here's what I want you
to do.
So here's the same universal
set, the same original sets I
had, E and F. I want you
to do E intersect F complement
and then E complement
intersect F complement.
So put the video on pause,
try this one
on your own first.
All right.
So I've got
to do E intersect F. So I have
to see what they have
in common.
Let's see.
What do they have in common?
I only see a 5.
So you have to write
that as a set.
Remember E intersect F is
a set.
So you need to put
that in braces, right?
And now we want
the complement.
Well, that means everything
in the universe
up here except 5.
So that's the numbers 1, 2, 3,
4 -- not 5 -- 6, 7, 8, and 9.
That's it.
So I have a little bit more
room for this one now.
And I'm going to need it.
So I'm going
to do E complement first.
Notice there's not --
I'm doing order
of operations here.
So I have to do E
complement first.
And we actually did
that in the previous problem.
Remember here's E complement
right here -- 2, 4, 6, 8, 9.
And here was F complement.
The only difference is I'm
going to change from a union
to an intersection, right?
So this first set is
E complement.
Second set is F complement.
And now we want
to take their intersection,
what these two sets have
in common.
Now let's see,
do they both have a 2?
Yes. Sorry.
All right.
Do they both have a 4?
No. Do they have a 6?
No. Do they have an 8?
Yes. Do they both have a 9?
Yes. So notice E intersection
F complement
over here we get these numbers
here, right?
Everything's at 5.
But E complement intersection
F complement is this.
So these are not equal,
either.
So just note E intersection F
complement is not the same
thing as E complement
intersection F complement.
All right.
Here's something cool
to notice.
We started
out with this universal set
and these two subsets E and F.
And we computed E union F
complement
and E complement union F
complement
and we realize those were not
the same.
So E union F complement was
not equal to E complement
union F complement.
Then we did E intersection F
complement
and we computed E complement
intersection F complement
and similarly we found
out those were not the same.
But there is something
interesting here.
Look at E union F complement,
right, and look
at E complement intersection
F complement.
They're the same.
So E union F complement ended
up being E complement
intersection F complement.
Similarly, E --
Eintersection F complement,
right, down here,
this is the same thing
as this --
E complement union
F complement.
So check it out.
You can't just, you know,
take the complement out
and put it in front and keep
that union.
What happens is it's sort
of like a
distributor property.
To get the rid
of the parenthesis,
it ends up being an E
complement and F complement
but whereas
in the parenthesis it was a
union outside the parenthesis
is an intersection.
And for the second one
when it was an intersection
in the parenthesis the other
one was a union.
So that's kind of interesting.
It's actually a distributive
property with sets that works.
[ Pause ]
>> Julie Harland:
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.