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