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PROFESSOR: So we've seen
that partial orders are

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a set of axioms that capture
the positive path relation

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or the arbitrary path relation
in directed acyclic graphs,

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or DAGs.

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But there's still another way
to understand these axioms

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that gives a kind
of representation

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theorem for the kind
of mathematical objects

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that are partial orders and that
every partial order looks like.

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So let's look at that example.

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I'm interested in the
proper subset relation.

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A is a proper subset of
B, which, you remember,

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means that B has everything
in it that A has and something

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extra.

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So in particular, since
B has something extra,

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B is not a subset
of A, certainly not

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a proper subset of A.

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So let's look at
an example of that.

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Here are seven
sets, and the arrows

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indicate the proper
subset relation.

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Or more precisely, the positive
path relation in this graph

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represents the proper
subset relation

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where edges are understood
to be pointing upwards.

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So I've left out the arrowheads.

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This is also known
as a Hasse diagram,

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where the height
is an indication

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of which way the arrows go.

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So if arrows are
pointing up, this

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is telling me that, for example,
this set of two elements,

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1 and 5, because there's
a path up to the top set,

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the top set has everything
that this lower set has.

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Namely the top set has 1 and
5, and it's got extra stuff.

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The set consisting of just 1
is a proper subset of 1 and 5

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because the set has 1 in it,
but it has an extra thing, 5.

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And also, there's a
path from 1 up to 1,

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2, 5, 10 because 1, 2, 5, 10
has a 1 in it and extra stuff.

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So that's what the
picture is illustrating,

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the proper subset relation
on this particular collection

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of seven sets.

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Now, let's look at a
very similar example

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of the proper divides
relation on some number.

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So proper divides
means a properly

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divides b if a divides b
and it's not equal to b.

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And I'm interested in the
proper divides relation

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on this set of seven numbers,
1, 2, 3, 5, 10, 15, and 20.

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And now there's a path from 5 to
30 because 5 is a divisor of 30

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and it's not equal to 30.

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It's a proper divisor of 30.

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And of course, the
point of this picture

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is to show that the proper
divides relation on these seven

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numbers has exactly the same
shape as the proper subset

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relation on those seven sets.

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So there's the seven sets and
their proper subset relation

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shown by the picture followed
by the proper divides relation

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on this set of seven numbers.

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And the precise notion
or sense in which

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these things have
the same shape--

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obviously they can be drawn and
one superimposed on the other.

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But abstractly what we care
about with partial orders

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and digraphs in
general is when things

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are isomorphic-- is
the technical name

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for the same shape--
and isomorphic means

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that all we care about
are the connections

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between corresponding vertices.

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Two graphs where the
vertices correspond in a way

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that where there's a
connection between two vertices

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there's also a connection
between the corresponding

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vertices are isomorphic.

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And the precise
definition of isomorphic

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is that they're
isomorphic when there's

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an edge-preserving matching
between their vertices.

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Matching means bijection.

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And the formal definition is G1
is isomorphic to G2 if and only

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if there's a bijection from
V1, the vertices of G1,

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to V2, the vertices of G2, with
the property that if there's

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an edge between two vertices u
and v in the first graph, then

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there's an edge between the
corresponding two vertices

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f of u and f of v
in the second graph.

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And that's an if and
only if relation.

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There's an edge between f
of u and f of v if and only

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if there's an edge between u
and v in the original graph.

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And that's the
official definition

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of isomorphism for digraphs.

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And the theorem
that we illustrated

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with that example of proper
divides and proper subset

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is that, in fact, every
strict partial order

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is isomorphic to some collection
of subsets partially ordered by

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less than.

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So this is a kind of a
representation theorem.

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If you want to know what kinds
of things are partial orders,

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the answer is that a
strict partial order

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is something that looks
like a bunch of sets

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under containment.

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It's isomorphic to a bunch
of sets under containment.

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And the proof, actually, of
this is quite straightforward.

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What I'm going to do to find
an isomorphism is you give me

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your arbitrary strict
partial order R,

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and I'm going to map an
element a in the domain of R

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to the set of all of
the elements that are,

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quote, "below it," that is,
all of the elements that

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are related to R.

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So a is going to map
to the set of b's such

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that bRa or b is equal to a.

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And that is added for
a technical condition.

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Remember, R restrict, so a
is not related to a under R.

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But I want it to be
in the set that a maps

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to, so I'm throwing that in.

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Another way to say this
is that the mapping f of a

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is equal to R
inverse of a union a.

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And let's just illustrate
that by the example of,

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how do you turn the divides
relation into the subset

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relation?

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Well, the smallest element
in the proper divides example

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was the number 1, and
I'm going to map it

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to the set consisting
of 1, which

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is all of the elements that
properly divide 1 along with 1.

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And then I'm going
to map the number

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3 to all of the elements that
properly divide 3 along with 3,

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and that is 1 and 3.

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5 maps to 1 and 5.

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2 maps to 1 and 2.

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And at the next level,
I'm going to map 15

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to all of the
numbers that properly

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divide 15 along with 15.

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So 1, 3, 5, and 15 are
what the number 15 maps to.

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That's a set.

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Likewise, 10 maps to 1,
2, 5, 15, and 30 maps

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to all of the numbers that are
below it, including itself.

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And this is the general
illustration of the way

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that you take an arbitrary
strict partial order and map

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elements into sets, which are
basically their inverse images

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under the relation.

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And the sets have exactly
the same structure

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under proper containment
as the relation.

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So this is, again, a
representation theorem

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that tells us that if we want
to understand partial orders,

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they are doing nothing more
than talking about relations

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with the same structure as
the proper subset relation

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on some collection of sets.