>> 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.
All right this is an
introduction to sets.
A set is simply a collection
or you could say it's a
collection of objects.
For instance,
a set could be a set
of stamps you had collected
or let's say you're doing some
math it might be some numbers;
those really don't look
like objects
but it's any collection you
have at all.
Now, the way we,
the notation we use for set is
by using braces.
In math, one way
to denote a set is
by using braces.
The objects
in the set are called elements
and we separate the elements
with commas.
So here is an example
of a set.
You have these braces.
So it's not a parentheses
and let's say I've got some
numbers in my set.
So let's say I have these
numbers, 2, 5, 6 and 8.
That's an example of a set.
It's a collection
of 4 numbers.
Sometimes we
like to call the set something
so we don't have to write
out the whole set
and usually we use a capital
letter but you don't have to.
I'm going to call this set A.
All right.
So this is the set A
and this is what set A
looks like.
It's got the braces,
4 numbers with commas.
Now another notation
that you sometimes use
that helps you tell how many
numbers are
in a set you put the number in
and a parentheses
and then you write the set.
So, you could either write A
or you could actually write
the whole thing
with the braces
and the numbers
and the commas.
I'm just going
to put the number of elements
in A. That N means the number
of elements in A. For those
of you who know function
notation like F of X,
that's not what this is,
all right?
So, it's in the context
of talking about the set A,
the number of elements
in A. So, if you go over here
and count them,
there are 4 elements.
Now it doesn't matter the
order that you write the
numbers in;
that's why we use braces
not parentheses.
Parentheses
when you have an ordered pair
it matters the first number is
the X coordinate,
the second number is the
Y coordinate.
Here it doesn't matter.
So I could have switched
and put the 6 here
and the 2 here;
it makes no difference the
order you write the numbers in
or whatever else happens to be
in your set.
Now, a lot
of times what we want
to do is sort
of restrict the universe
and what we're talking
about to something
in particular
and that's called the
universal set.
So let's call it and it's
like a script capital U. Let's
say the universal set are
these numbers, 1, 2, 3, 4, 5,
6, 7, 8, 9.
That's it.
Let's say that's the universe.
We're not going
to consider any numbers except
1 through 9.
So, A of these numbers has 2,
5, 6, and 8.
Let's make another set
from the universal set.
So, if I pick some other
numbers, how
about if I pick 3,
4 and 5, okay?
That's another set.
Let's pick another set;
how about 5, 8 and 9?
These are all subsets; A,
B and C are called subsets
of the universal set.
So, a subset has got either
the same amount of elements
or fewer elements
than the bigger set
and the notation we use
for subset looks like this.
This line under here means it
could be equal.
If you don't have this little
line under here,
it means it can't be exactly
the same set.
So, in our example,
A is a subset
of the universal set;
B is a subset
of the universal set;
and C is a subset
of the universal set.
That's how it's always
supposed to be.
If you're going
to pick some sets
from the universal set,
it has to be that any
of the elements in A have
to be in the universal set.
So, B, these elements are
chosen from the universal set.
I could also have a subset
of A. So, if we look at A,
let's pick a subset
from there.
How about 5 and 6?
So we'll take the set 5 and 6
and I'm going
to say that's a subset of A
because it's a set
that has some of the elements
that are in A. Now,
what about this,
what about if I have 3, 4,
5 can I say that's a subset
of B?
Sure. Because it contains
elements, only elements
that are in B. So it comes
from the elements of B
and that's why this line
is important.
It could actually equal B
as well.
If you take that line off,
what it's saying is it cannot
be equal.
So it can either be 3
or just the number 4
or just number 5 or 3 and 4
or 4 and 5 or 3 and 5
but not all the numbers;
it can't exactly be equal.
All right let's take this same
universal set here
and we've got 3 subsets
of that, A,
B and C. Here's the notation.
If you have this kind
of long-looking C thing
with an underline,
it means is a subset of
and it means really it could
be a proper subset
or it can actually be equal
and it would
at the equal sign means is a
proper subset of,
which means it has
to have fewer elements
from the original one.
So, let's decide if each
of these statements I have
down here are true or false.
The set 6, 8 is a subset of A.
So what you have to do is say
as long as all the elements
in here are also in A
and there's fewer elements
then it's going to be true.
So, what you're looking
for to say it's false is
a contradiction.
Like if it had the number 1
in it, that would be
a contradiction.
So, 6 and 8 is
that set a subset of A?
Yes. So that's true.
Next one, the set that has 3
and 6 is that a subset of B?
So make sure that any elements
in here are also in B. Well,
3 is in but 6 is not
so that's false.
Five, 8 and 9 that set with 5,
8 and 9 is that a set,
a proper subset?
I haven't been saying this,
is it a proper subset,
right, of C?
So you look.
No, it's not
because it's actually equal
to C so this one is false.
If it's exactly the same,
to make it true you would have
to have that line under here.
Without the line it's false.
Three, 4 and 5 is
that a subset of B?
It's actually equal to B
so the answer is going
to be true.
I don't have room
to write true.
There we go.
How about the set
with the number 9 in it?
Is that a subset of C?
Yes, because 9 is in there.
Here's the trickiest one
of all the last one.
What's in there?
There are no numbers in there.
So, there can't be
any contradiction.
So actually the set
that has nothing in it,
nothing at all in it,
so there's no elements is
going to be a subset.
It doesn't matter what it is,
it doesn't matter if I put A,
B, C, D, F, G,
U whatever it is the answer is
always going to be true.
That's hard for people
to understand,
but this is called the
empty set.
The empty set is always
a subset.
So in other words, to make one
of these statements false,
you have to give an example
of something in the set
that is not in the bigger set,
the set on the right
hand side.
So, for instance,
this second one here
where I have 3
and 6 is a subset of B,
my contradiction is
right here.
I could tell them, hey,
6 is not in B
and that's the reason
it's false.
You have to have a reason
like that.
All right.
So this set with no elements
in it is called the empty set
or null set;
that's how we usually refer
to it in math and another way
to write the null set it looks
like a 0 with a line
through it.
So those of you that are used
to writing the 0 as a 0
with a line through it,
it's going to be confusing
in math because we actually
use that to represent the
null set.
So here's a tricky thing.
Is that the null set?
No. Because here's the
weird part.
It's a set
and now I see there's
something in it.
I know there's something
in the null set
but that's something.
So this is not the null set.
It's either you write the
braces with nothing in it
or you write that 0
if you want to think of it or
and oh with that diagonal line
through it.
There's a lot more to sets
than this and sets can be
described in words,
sets can have infinitely many
elements in it, et cetera,
so here are just a couple
more examples.
I could say the set
of US Presidents served
in the 1900s
and then I would just list all
the presidents from the 1900s.
Well, that's a long list,
I'm not going
to do it right here,
but you would begin writing
the first person's name
and then put a comma
and the next person's name,
et cetera,
until you listed them all.
So that's one way of doing it,
but at least there's only a
certain number of presidents
that served in the 1900s.
How about the set
of even whole numbers?
There's a problem here.
It's impossible to list all
of these, but there is a way
of showing what you mean.
The first even whole number,
actually the first even one is
0 because I said a whole
number, 0 is a whole number,
0, 2, 4, and of course,
I can't list all of them,
it's absolutely impossible
because this goes on forever.
So, if something goes
on forever, what you do is
after one of the commas you
establish a pattern
so people could get the idea
what's going on
and then you put 3 dots.
This means and so on,
this just keeps going
on forever and ever.
All right so I'm not going to,
you know, discuss too much
more because there's just a
lot of different ways
that you could describe sets
and lots of different
examples, but in math mainly
we're going
to probably use maybe we'll
have a set
with maybe some numbers
or some letters in it usually
and so in the next video I'm
going to work
with sets a little bit more,
you know, after you have a set
what can you do with them,
what are some other symbols we
use in set theory.
[ Silence ]
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.