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PROFESSOR: We're going
to look at the most

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fundamental of all mathematical
data types, namely sets,

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and let's begin with
the definitions.

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So informally, a
set is a collection

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of mathematical
objects, and the idea

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is that you treat the collection
of objects as one new object.

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And that's a working definition.

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But of course, it might
help a little bit,

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but it's a circular definition.

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This is not math yet
because I haven't defined

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what a collection
is, and a collection

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is no clearer or easier
to define than a set is.

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So let's try to work
up the idea of sets

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by looking at some examples.

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So we've already talked
about some familiar ones.

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There's the real
numbers for which

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we had this symbol
R in a special font,

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and the complex numbers
C, and the integers

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Z. And we might have
mentioned, and you

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might have seen already, the
idea of the empty set for which

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we use this symbol that
looks like a zero with a line

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through it.

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Let's look at an example
to pin things down.

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Let's look at this
set of four things.

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Namely, it's got two numbers,
pi/2 and 7, a character

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string in quotes, "Albert R,"
and the Boolean value true.

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So those are the four
different things in it.

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They're of mixed type.

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And you might not like to
have a mixed type like this

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in a programming language.

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But mathematicians don't worry
about such things very much,

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rarely.

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Anyway, the first
observation is that the order

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in which these elements
are listed doesn't matter.

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This set-- the braces
indicates that it's

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the set of these
things-- is the same if I

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listed T first, then the string,
and the two numbers last.

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There is no notion
of order in a set.

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Now, to a computer scientist
this is a little unnatural.

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The most natural
thing to be would

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be to define a
sequence of things,

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like the sequence that
began with 7, then

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had the character string,
then had the number,

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then had the Boolean.

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And you could get by with
working with lists of things

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as long as they're finite.

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But they very quickly
get out of hand

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when you have to talk
about, say, a set of lists.

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Then it's not clear how to
make a list out of those,

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and you wind up
making sets again.

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So sets, in fact, are an
unavoidable kind of idea.

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So another basic
thing to understand

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about the notion of a
set is that an element

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is either in a set
or not in a set.

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So if I write down
7, pi/2, 7, this

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is the same description
of the same set 7, pi/2.

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I'm just telling you the
same thing twice here.

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That 7 is in the set, and
the 7 is in the set again.

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So no notion of being in
the set more than once.

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Now, sometimes,
technically you want

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to add a notion of
so-called multisets

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in which elements
can be in a set

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a certain number of times,
an integer number of times.

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But there's no
real need for that.

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It's a secondary idea.

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And from our point of view,
you're in or out of a set.

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If you repeat elements, it's the
same as mentioning them once.

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So the most fundamental feature
of a set is what's in it.

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And for that, there's
a special notation.

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So we'll say that x is a
member of A, where A is a set,

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and use this epsilon symbol
to indicate membership.

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It's read x is a member
of A. So for example, pi/2

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is a member of that set that we
saw before that had pi/2 in it.

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14/2 is also a member of
that set because 14/2 is just

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another description of 7.

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When I write 7 here, I
don't mean the character 7.

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I mean the number 7.

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And so 14/2 is the description
of the same number.

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It's in that set.

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On the other hand,
pi/3 is a number

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that's simply not in that set.

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So I'm using the epsilon with
a vertical bar through it,

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or some kind of a line through
it, to mean not a member of.

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Membership is so basic that
there's a lot of different ways

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to say it.

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Besides using the membership
symbol x is a member of A,

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you can sometimes say
x is an element of A,

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or x is in A, as well
as x is a member of A.

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They're all synonyms.

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So for example, 7 is a
member of the integers.

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Z is our symbol
for the integers.

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2/3 is not a member
of the integers

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because it's a fraction
that's not an integer.

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And on the other hand,
the set Z of integers

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itself is a member of this
three-element set consisting

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of the truth value T,
the set of all integers,

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and the element 7.

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So here's an example where a
set can contain sets, quite

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big ones even, and that's fine.

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That's not any problem
mathematically.

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Related to membership is another
fundamental notion of subset.

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So A is a subset of
B, it's pronounced.

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So that horizontal U
with a line under it

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is meant to resemble a less
than or equal to symbol.

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So you can think of it as being
A is less than or equal to B.

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But don't overload the symbols.

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Less than or equal to is used
on numbers and other things

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that we know how to order.

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And this is a relation that's
only allowed between sets.

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So A is a subset of B-- A
synonym is that A is contained

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in B-- simply means
that every element of A

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is also an element
of B. If I wrote

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that out in predicate
logic notation

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as a predicate formula, I'd say
for every x, x is in A implies

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x is in B. If it's in A, then
it's in B. Everything in A

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is in B.

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So some examples of
the subset relation

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are that the integers are
a kind of-- an integer

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is a special case
of a real number.

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So the set of integers is a
subset of the real numbers.

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A real number is a special
case of a complex number,

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so the real numbers are a
subset of the complex numbers.

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And here's a concrete
example, where

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I have a set of three
things, 5, 7, and 3,

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and this is the set with
just the element 3 in it.

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Now, we sometimes are
sloppy about distinguishing

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the element 3 from the set
that's consisting of just 3

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as its only element.

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But in fact, it's a pretty
important distinction

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to keep track of.

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In this case, 3 is not a subset
of this set on the right.

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But the set consisting of 3 is
a subset of the set on the right

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because, after all, the only
member of this set is 3,

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and that is a
member of this set.

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A consequence of this general
definition is that every set is

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a subset of itself because
everything in A is in A .

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That's not really
very interesting.

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Another important
general observation

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is that the empty set is
a subset of everything.

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The empty set is a
subset of every set.

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Let's look at why that
is in more detail.

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So the claim is that the empty
set is a subset of everything.

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Let B be any old set,
then the empty set

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is a subset of B. What exactly
does that mean according

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to the definition of subset?

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Well, it says that everything
that's in the empty set,

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if it's in the
empty set, then it

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implies that it's in
B. For every element,

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if it's in the empty
set, then it's in B.

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Well, what do we
know about this?

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The assertion that x is
in the empty set is false.

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No matter what x is, there's
nothing in the empty set.

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And now I have an
implication that

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implies where the left-hand
side, the hypothesis, is false.

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That means that the whole
implication is true,

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and it doesn't
depend on what B is.

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I'm not even going
to look at B. I

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can see that x is in
empty set is false,

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so the whole
implication is true.

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And so what I'm saying
is that for every x,

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something that's
true has to be true.

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Well, it is.

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And that's why the empty
set is a subset of B

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satisfies this definition
in a formal way.

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And this is an example
of why that convention

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that false implies anything
is convenient and is made

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use here.

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So when you're defining
sets, if they're small,

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you can just list
the elements, as we

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did with that set with 7
and pi/2 and "Albert R."

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Sometimes we can even
describe infinite sets

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as some kind of a list.

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Like I might describe the
set of integers as saying,

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well, it's 0, 1, minus
1, 2, minus 2, and so on,

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and you would understand that.

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But in general,
if I'm describing

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a set that is not
so easy to list,

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say the real numbers,
then what I'm going to do

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is define a set by a defining
property of in the set.

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So I'm interested in a
property P of elements,

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and I'm going to look
at the set of elements

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x that are in some set A
such that P of x is true,

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and that's the notation we use.

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So this would be read as the
set of x in A such that P of x

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holds, that x has property P.
So notice this vertical bar is

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read as "such that."

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It's just a mathematical
abbreviation.

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This is those elements in A
that have property P that P of x

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holds for, and that defines
a set of those elements.

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Let's look at a simple example.

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The set E of even integers is
simply the set of numbers n

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that are integers
such that n is even.

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So in this case, the property
P of n means that n is even.

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One last concept is the
concept of the power set.

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So the power set of a set A
is all of the subsets of A.

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So we could define
it using set notation

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as it's the set of
B such that B is

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a subset of A. An
example would be--

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let's take the power set
of the two Boolean values

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true and false.

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So the power set
of true and false,

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of that set consisting
of two elements,

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is-- well, what are
some of its subsets?

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The set consisting of just true
is a subset of true, false.

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So is the set
consisting of false,

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and so is the whole thing.

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It's a subset of itself.

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And one final element,
the empty set,

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is a subset of the set of
Boolean values true and false.

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So the power set of
this two-element set

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is a set that has four things
in it-- two elements of size 1,

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one element of size 2,
one element of size 0.

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00:10:53,480 --> 00:10:55,860
And that's going to be
a general phenomenon

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that we'll examine more later.

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How big is the
power set of a set?

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The even numbers, E that we just
defined on the previous slide,

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is a member of the power
set of Z because it's

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a subset of integers.

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Even integers are a
special case of integers.

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And the integers are a
member of the power set of R.

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That's just a synonym
for saying that integers

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are a subset of reals.

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Every integer is a
real, so the integers

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are a subset of
reals, which means

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they're a member of
the power set of reals.

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So the general property
is that a set B

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is a member of the power
set of A if and only

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if B is a subset of A.
That was the defining

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condition for power set.

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And that's a fact to remember,
and it may potentially

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confuse you.

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00:11:50,500 --> 00:11:53,720
But it's a good
exercise in keeping

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track of the difference
between is a member of

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and is a subset of.