In this segment, we will
talk about sets.
I'm pretty sure that most of
what I will say is material
that you have seen before.
Nevertheless, it is useful to
do a review of some of the
concepts, the definitions, and
also of the notation that we
will be using.
So what is a set?
A set is just a collection
of distinct elements.
So we have some elements, and
we put them together.
And this collection, we
call it the set S.
More formally, how do
we specify a set?
We could specify a set by
listing its elements, and
putting them inside braces.
So this is a set that consists
of four elements, the letters,
a, b, c, d.
Another set could be the set
of all real numbers.
Notice a distinction here--
the first set is a finite set.
It has a finite number of
elements, whereas the second
set is infinite.
And in general, sets are
of these two kinds.
Either they're finite,
or their infinite.
A piece of notation now.
We use this notation to indicate
that a certain object
x is an element of a set S. We
read that as x belongs to S.
If x is not an element of S,
then we use this notation to
indicate it, and we read it
as x does not belong to S.
Now, one way of specifying
sets is as follows.
We start with a bigger
set-- for example,
the set of real numbers--
and we consider all of those
x's that belong to that big
set that have a certain
property.
For example, that the cosine of
this number is, let's say,
bigger than 1/2.
This is a way of specifying
a set.
We start with a big set, but
we then restrict to those
elements of that set that
satisfy a particular property.
One set of particular interest
is the following.
Sometimes in some context, we
want to fix a collection of
all possible objects that we
might ever want to consider,
and that collection
will be a set.
We denote it usually by
omega, and we call it
the universal set.
So having fixed a universal
set, we will only consider
smaller sets that lie inside
that big universal set.
And once we have a universal
set, we can talk about the
collection of all objects, or
elements, that belong to our
universal set, but do not belong
to the set S. So that
would be everything
outside the set S.
Everything outside the set S, we
denote it this way, and we
call it the complement of the
set S. And it is defined
formally as follows--
an element belongs to the
complement of S if x is an
element of our universal set,
and also x does not belong to
S. Notice that if we take the
complement of the complement--
that is, anything that does not
belong to the green set--
we get back the red set.
So what this is saying is that
the complement of the
complement of a set
is the set itself.
Another set of particular
interest is the
so-called empty set.
The empty set is a set that
contains no elements.
In particular, if we take
the complement of
the universal set--
well, since the universal set
contains everything, there is
nothing in its complement, so
its complement is going to be
the empty set.
Finally, one more piece
of notation.
Suppose that we have two
sets, and one set is
bigger than the other.
So S is the small set here,
and T is the bigger set.
We denote this relation by
writing this expression, which
we read as follows--
S is a subset of the set T.
And what that means is that if
x is an element of S, then
such an x must be also an
element of T. Note that when S
is a subset of T, there is also
the possibility that S is
equal to T. One word
of caution here--
the notation that we're using
here is the same as what in
some textbooks is written
this way--
that is, S is a subset of T, but
can also be equal to T. We
do not use this notation, but
that's how we understand it.
That is, we allow for the
possibility that the subset is
equal to the larger set.
Now when we have two sets, we
can talk about their union and
their intersection.
Let's say that this is set S,
and this is set T. The union
of the two sets consists of all
elements that belong to
one set or the other,
or in both.
The union is denoted this way,
and the formal definition is
that some element belongs to the
union if and only if this
element belongs to one of the
sets, or it belongs to the
other one of the sets.
We can also form the
intersection of two sets,
which we denote this way,
and which stands for the
collection of elements that
belong to both of the sets.
So formally, an element belongs
to the intersection of
two sets if and only
if that element
belongs to both of them.
So x must be an element of S,
and it must also be an element
of T.
By the way, we can also define
unions and intersections of
more than two sets, even of
infinitely many sets.
So suppose that we have an
infinite collection of sets.
Let's denote them by Sn.
So n ranges over, let's say, all
of the positive integers.
So pictorially, you might
think of having one set,
another set, a third set, a
fourth set, and so on, and we
have an infinite collection
of such sets.
Given this infinite collection,
we can still
define their union to be the
set of all elements that
belong to one of those sets
Sn that we started with.
That is, an element is going to
belong to that union if and
only if this element belongs
to some of the sets that we
started with.
We can also define the
intersection of an infinite
collection of sets.
We say that an element x belongs
to the intersection of
all these sets if and only if
x belongs to Sn for all n.
So if x belongs to each one of
those Sn's, then we say that x
belongs to their intersection.
Set operations satisfy certain
basic properties.
One of these we already
discussed.
This property, for example,
is pretty clear.
The union of a set with another
set is the same as the
union if you consider the two
sets in different orders.
If you take the union of three
sets, you can do it by
forming, first, the union of
these two sets, and then the
union with this one; or, do it
in any alternative order.
Both expressions are equal.
Because of this, we do not
really need the parentheses,
and we often write just this
expression here, which is the
same as this one.
And the same would be true
for intersections.
That is, the intersection of
three sets is the same no
matter how you put parentheses
around the different sets.
Now if you take a union of a set
with a universal set, you
cannot get anything bigger than
the universal set, so you
just get the universal set.
On the other hand, if you take
the intersection of a set with
the universal set, what is left
is just the set itself.
Perhaps the more complicated
properties out of this list is
this one and this one, which
are sort of a distributive
property of intersections
and unions.
And I will let you convince
yourselves
that these are true.
The way that you verify them
is by proceeding logically.
If x is an element of this, then
x must be an element of
S, and it must also be an
element of either T or U.
Therefore, it's going to belong
either to this set--
it belongs to S, and it also
belongs to T-- or it's going
to be an element of that set--
it belongs to S, and it
belongs to U.
So this argument shows that
this set here is a
subset of that set.
Anything that belongs
here belongs there.
Then you need to reverse the
argument to convince yourself
that anything that belongs here
belongs also to the first
set, and therefore, the
two sets are equal.
Here, I'm using the following
fact-- that if S is a subset
of T, and T is a subset of
S, this implies that the
two sets are equal.
And then you can use a similar
argument to convince
yourselves about this
equality, as well.
So this is it about basic
properties of sets.
We will be using some of these
properties all of the time
without making any special
comment about them.
