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PROFESSOR: We've seen
a lot of functions
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in introductory calculus-- trig
functions, rational functions,
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exponentials, logs and so on.
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I don't know whether
your calculus course
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has included a general
discussion of functions.
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The old fashioned ones didn't,
and we will go into that
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now in this segment.
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And we're going to be
interpreting functions
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as a special case
of binary relations.
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So let's just say what
a binary relation is.
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A binary relation is
a mathematical object
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that associates
elements of one set
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called the domain with
elements of another set
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called the codomain.
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And I'm going to give
you a bunch of examples
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of binary relations
in a short moment,
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but let's just talk
about what they're for
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and what their role is.
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So they may be familiar to
you as computer scientists
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if you've worked with any
relational databases like SQL
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or MySQL.
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MySQL.
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And we'll see an
example that indicates
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where the original ideas behind
those relational databases
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came from.
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But even more
fundamental, relations
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are one of the most basic
mathematical abstractions right
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after sets, and they
play a role everywhere.
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We're going to be
looking in later lectures
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at special kinds
of binary relations
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like equivalence relations
and partial orders
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and numerical congruences.
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But today, we're going
to set up the machinery
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to be using binary
relations for counting,
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which will be another important
application in this class.
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So let's look at an example.
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And I'm going to take one that's
close to home-- the registered
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for relation, which is a
relation between students--
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that's going to be the
domain, in this case, four
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students, Jason, Joan, Yihui,
and Adam-- and four subjects.
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As a coincidence, 6.042,
003, 012, and 004.
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And the relation R is
going to be indicated
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by arrows which show just
which students are associated
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with which subjects,
meaning that they're
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registered for that subject.
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So if we look at
Jason, we can see
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that there's a particular arrow
connecting Jason and 6.042.
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And what that tells us is that
Jason is registered for 6.042.
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Now, there's a
bunch of notations
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that are used with respect
to binary relations.
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So let's look at some.
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One way to write
it is if you think
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of the relation R
as an equality sign
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or a less than sign,
where it's normally
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written in the middle
of the two things
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that it's connecting, as in
this example-- Jason R 6.042.
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That would be called
infix notation.
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Sometimes it's written
as a binary predicate-- R
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of Jason comma 6.042.
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That would be kind
of prefix notation
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where the relation or
operator comes first.
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And then if you start
being a little closer
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to the formal definition of
what a binary relation is,
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you could say that the
ordered pair Jason 6.042
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is a member of the relation.
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If you wanted to
be really precise,
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you would say that
it was a member
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of the graph of the relation.
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And I'll come back
and elaborate further
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on what the graph
of a relation is
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and what this ordered
pairs businesses.
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But for now, just let's
continue with this example.
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And a basic concept
with relations
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is the idea of the image
of a bunch of domain
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elements under the relation.
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So you can think of the
relation as an operator
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that applies to domain
elements or even sets
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of domain elements.
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So if I write R of Jason,
that defines the subjects
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that Jason is registered for.
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So looking at the picture,
R is not a function.
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So that there may be more
than one subject, as is
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you'd expect for a student to be
registered for multiple courses
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at MIT.
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So Jason in this diagram is
registered for 6.042 and 6.012
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as indicated by the
highlighted two connection
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arrows, which we've made red.
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Which means that R
of Jason is that set
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of two courses that
he's associated with
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or that are associated
with him-- that he's
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registered 6.042 and 6.012.
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So at this point,
we've applied R
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to one domain element--
one student Jason.
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But the interesting
case is when you apply R
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to a bunch of students.
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So the general
setup is that if x
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is a set of students-- a subset
of the domain, which we've
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been showing in green--
then if I apply R to X,
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it gives me all the
subjects that they're
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taking among them-- all the
subjects that any one of them
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is taking.
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Let's take a look at an example.
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Well, another way to say
it I guess is that R of X
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is everything in R that
relates to things in X.
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So if I look at
Jason and Yihui and I
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want to know what do
they connect to under R--
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these are the subjects
that Jason or Yihui
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is registered for.
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The way I'd find that is by
looking at the arrow diagram,
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and I'd find that Jason
is taking 042 and 012.
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And Yihui is taking 012 and 004.
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So between them, they're
taking three courses.
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So R of Jason, Yihui is
in fact 042, 012, and 004.
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So another way to understand
this idea of the image of a set
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R of X is that X is a
set of points in the set
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that you're starting
with called the domain.
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And R of X is going to be all of
the endpoints in the other set,
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the codomain, that start at X.
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If I said that as a statement
in formal logic or in set theory
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with logical notation,
I would say that R of X
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is the set of j in
subjects such that there
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is a d in X such that dRj.
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So what that's exactly
saying that dRj
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says that d is the starting
point in the domain.
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d is a student.
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j is a subject.
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dRj means there's an arrow
that goes from student
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d to subject j.
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And we're collecting the set of
those j's that started some d.
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So an arrow from X
goes to j is what
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exists at d an X. dRj means--
written in logic notation--
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it's really talking about
the endpoints of arrows,
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and that's a nice way
to think about it.
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But you ought to be
able also to retreat
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to give a nice, crisp
set theoretic definition
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without reference to
pictures if need be.
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So that's an official
definition of the image under R.
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Let's turn now to an
operation on relations which
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converts one relation
into another relation
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called the inverse of
R. And the inverse of R
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is what you get by
turning the arrows around.
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So let's look at the
relation R, which
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is the registered for
relation going from d students
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to j subjects.
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And then if I look at
R inverse, R inverse
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I could think of as the
registers relation-- 6.042
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registers Jason, and 6.012
registers Jason and Yihui.
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It's a funny usage of the word,
but I needed something short
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that would fit on the slide.
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So registers is basically
turning the arrows backwards of
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is registered for.
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And now I can apply
the definition of image
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to R inverse in a useful way.
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But just to be crisp about
what we're doing here
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is formally our
inverse is gotten
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by flipping the role of the
domain and the codomain.
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So we have that dRj if
and only if jR inverse d.
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So let's look at R
inverse of 6.012.
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What that's going to
mean is all the students
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that are taking 6.012.
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So we start off at 6.012, and
we go back to all the students
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that are registered for it.
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It's Jason and Yihui again.
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And so our inverse of
6.012 is Jason and Yihui.
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Our inverse of 6.012 and 6.003?
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Well, it's same deal.
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Let's look at 6.003 and 6.012
and look at all the students
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that are registered
for either one of them.
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Now its Jason, Joan, and Yihui
shown by those red arrows--
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all the arrows coming out of
those two courses, 003 and 012.
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And so our inverse of 003
and 012 is that set of three
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students-- Jason,
Joan, and Yihui.
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And in general,
when you start off
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with a bunch of subjects--
a bunch of elements--
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of the codomain and you
apply R inverse to it,
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it's called the inverse
image of the Y under R.
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Well, let's look at the
set J of all the subjects
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and think about what
is R inverse of J. What
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does it mean?
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Well, R inverse of J
is all the students
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that are registered
for some subject
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at all, which is a
good thing to have.
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So now, I can start using
these sets to make assertions
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about my database that
can be useful to know.
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So for example, if I want
to say that every student is
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registered for some
subject-- which, of course,
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they are-- what I
would say is that D,
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the set of all students, is
a subset of R inverse of J.
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So this concise set theoretic
containment statement--
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d is a subset of
R inverse of J--
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is a slick way of writing
the precise statement that
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says that all the students are
registered for some subject.
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Now, happens not to
be true by the way.
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Because if you look
back at that example,
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Adam was not registered
for a subject.
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So we're not claiming
that this is true,
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but simply that
there's a nice way
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to express it using
images and containment.
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Let's look at a
different relation
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that we could call
the advises relation.
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So the advises relation's
going to have codomain
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the same set of students d, but
it's going to have as a domain
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the set of professors.
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And I've written down
the initials of five
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prominent professors
minus at the top--
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and you may recognize
some of the others.
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But it doesn't really
matter if you don't.
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And the advises
relation V is going
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to be indicated by those arrows.
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So in particular,
it shows that ARM
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is the adviser of Jason,
Joan, Yihui, and Adam,
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which he happens to be.
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FTL is an adviser
of Joan and Yihui.
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So Joan has two advisers
because she's a double major.
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Yihui does as well.
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And Adam does as well now
that I look at this example.
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So if I look at in particular
now the advisees of FTL or TLP,
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I'm looking at V of the set
consisting of FTL and TLP.
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And it's going to be
Joan, Yihui, and Adam.
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So taking the image
of FTL and TLP--
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that's the set of advisees of
either of those two professors,
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I get this set of
three students--
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Joan, Yihui, and Adam.
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Well, that's a set of students,
and the registered relation
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applies to a set of students.
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So let's do that.
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If I now apply R
to Joan and Yihui
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and Adam, what I'm
getting is the subjects
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that they're registered for.
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So this is called composing R
and V. I've applied V and them
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I'm applying R to the result.
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In this case, R of
V of FTL and TLP
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is the same as R of
Joan, Yihui, and Adam.
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It's the courses that any of
them are taking, and it's 003,
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012, and 004.
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So the way to understand
this R of V in general
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is you start off with any set
X of professors in the domain.
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You take V of W-- are the
advisees that they have have--
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and then you take
R of the advisees,
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and you get the subjects
that the advisees are taking.
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So R of V of X is the subjects
that advisees of X are taking,
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are registered for.
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00:13:33,590 --> 00:13:35,780
Well, we can abstract
that out and call
244
00:13:35,780 --> 00:13:38,840
this the composition of R and
V. It's defined the same way
245
00:13:38,840 --> 00:13:40,610
that functional composition is.
246
00:13:40,610 --> 00:13:45,690
So R of V is the relation and
the images of that relation.
247
00:13:45,690 --> 00:13:48,220
The images of a set of
professors under R of V
248
00:13:48,220 --> 00:13:55,550
is defined to be apply V to
X and then apply R to V of X.
249
00:13:55,550 --> 00:14:00,160
And it's again, called the
composition of R and V.
250
00:14:00,160 --> 00:14:05,460
What it means now is that two
things are related by R of V.
251
00:14:05,460 --> 00:14:08,790
It relates professors
and subjects.
252
00:14:08,790 --> 00:14:10,910
And it says that a
professor in a subject
253
00:14:10,910 --> 00:14:15,070
are related if the professor
has an advisee-- some advisee--
254
00:14:15,070 --> 00:14:17,700
in that subject.
255
00:14:17,700 --> 00:14:20,420
p for a professor.
256
00:14:20,420 --> 00:14:23,610
Composition of R with
V. j for a subject
257
00:14:23,610 --> 00:14:27,770
holds if and only if
professor p has an advisee
258
00:14:27,770 --> 00:14:30,650
registered in subject j.
259
00:14:30,650 --> 00:14:32,650
Let's see how you figure
that kind of thing out.
260
00:14:32,650 --> 00:14:36,010
So I'm going to draw
the V relation which
261
00:14:36,010 --> 00:14:39,380
goes from p professors
to D students
262
00:14:39,380 --> 00:14:42,220
and then the R relation
that goes from D students
263
00:14:42,220 --> 00:14:44,350
to J subjects.
264
00:14:44,350 --> 00:14:47,410
And by showing
them in this way, I
265
00:14:47,410 --> 00:14:50,930
can understand the
composition of R and V
266
00:14:50,930 --> 00:14:53,210
as following two arrows.
267
00:14:53,210 --> 00:14:58,430
You start off, say, at ARM, and
you follow a V arrow from ARM
268
00:14:58,430 --> 00:15:01,140
to his advisee, Yihui.
269
00:15:01,140 --> 00:15:06,050
Then you follow another
arrow from Yihui to 6.012,
270
00:15:06,050 --> 00:15:11,900
and you discover, hey,
ARM has an advisee in--
271
00:15:11,900 --> 00:15:15,910
So now we can say that
professor ARM is in the relation
272
00:15:15,910 --> 00:15:21,890
R composed with V with
6.012 because of this path
273
00:15:21,890 --> 00:15:27,490
ARM has Yihui as an advisee, and
Yihui is registered for 6.012.
274
00:15:27,490 --> 00:15:30,170
And this relation R
o V, we figured out,
275
00:15:30,170 --> 00:15:31,910
is the relation
that the professor
276
00:15:31,910 --> 00:15:34,750
has an advisee in the subject.
277
00:15:34,750 --> 00:15:36,740
So in general,
what we can say is
278
00:15:36,740 --> 00:15:39,820
that a professor p is
in the R o V relation
279
00:15:39,820 --> 00:15:42,290
to j if and only
if-- and here we're
280
00:15:42,290 --> 00:15:45,540
going to state it in formal
logical notation, which
281
00:15:45,540 --> 00:15:47,950
really applies in
general, not just
282
00:15:47,950 --> 00:15:50,130
to this particular example.
283
00:15:50,130 --> 00:15:52,720
So the definition
of R composed with V
284
00:15:52,720 --> 00:15:56,270
is the p R composed
with Vj means
285
00:15:56,270 --> 00:16:00,610
there's a D that connects
p and j through V and D,
286
00:16:00,610 --> 00:16:06,060
in particular that
there's a D such that pVd,
287
00:16:06,060 --> 00:16:09,172
which means there's a
V arrow from p to to d.
288
00:16:09,172 --> 00:16:13,630
And dRJ-- there's an
R arrow from d to j.
289
00:16:13,630 --> 00:16:16,308
For some, d.
290
00:16:16,308 --> 00:16:18,560
By the way, there's
a technicality here
291
00:16:18,560 --> 00:16:25,960
that when you write the
formula pVd and dRj, following
292
00:16:25,960 --> 00:16:28,230
the diagram where you
start with V on the left
293
00:16:28,230 --> 00:16:30,600
and follow a V arrow
and then and R arrow,
294
00:16:30,600 --> 00:16:33,080
it's natural to think
of them as written
295
00:16:33,080 --> 00:16:36,220
in left to right order of
which you apply first V R.
296
00:16:36,220 --> 00:16:38,830
But of course, that's not
the way composition works.
297
00:16:38,830 --> 00:16:42,940
When you apply, one function--
R to V to something,
298
00:16:42,940 --> 00:16:44,640
you're applying V first.
299
00:16:44,640 --> 00:16:46,210
And you write it on the right.
300
00:16:46,210 --> 00:16:50,840
So R o V is written like
function composition
301
00:16:50,840 --> 00:16:55,530
where V applies first,
but the logical statement,
302
00:16:55,530 --> 00:16:57,260
the natural way to
write it, is to follow
303
00:16:57,260 --> 00:16:58,970
the way the picture works.
304
00:16:58,970 --> 00:17:01,400
And D, Vs, and Rs get reversed.
305
00:17:01,400 --> 00:17:05,088
So watch out for that confusion.
306
00:17:05,088 --> 00:17:08,119
Well, I want to introduce
one more relation to flesh
307
00:17:08,119 --> 00:17:12,290
out this example, and that'll
be the teaches relation.
308
00:17:12,290 --> 00:17:14,300
So the teaches relation
is going to have--
309
00:17:14,300 --> 00:17:19,030
as domain professors, again--
and it's codomain, subjects.
310
00:17:19,030 --> 00:17:21,849
And it's simply going to
tell us who's teaching what.
311
00:17:21,849 --> 00:17:27,060
So here we're going to see
that ARM is teaching 6.042,
312
00:17:27,060 --> 00:17:28,210
as you well know.
313
00:17:28,210 --> 00:17:32,330
And FTL is teaching 6.042,
two which he does frequently
314
00:17:32,330 --> 00:17:34,890
but not this term.
315
00:17:34,890 --> 00:17:39,530
And now I can again use
my relational operators
316
00:17:39,530 --> 00:17:43,650
to start making assertions
about these people
317
00:17:43,650 --> 00:17:47,680
and relations involving
teaching and advisees.
318
00:17:47,680 --> 00:17:50,390
And a useful way to do that
is by applying set operations
319
00:17:50,390 --> 00:17:54,690
to the relations because I
can think of the relations
320
00:17:54,690 --> 00:17:56,700
as being that set of arrows.
321
00:17:56,700 --> 00:18:00,290
So suppose I wanted to make
some statement that a professor
322
00:18:00,290 --> 00:18:05,660
should not teach their
own advisee because it's
323
00:18:05,660 --> 00:18:09,992
too much power for one person
to have over a student.
324
00:18:09,992 --> 00:18:11,200
This is not true, by the way.
325
00:18:11,200 --> 00:18:13,830
It's common for professors
to teach advisees,
326
00:18:13,830 --> 00:18:15,740
but there are other
kinds of rules
327
00:18:15,740 --> 00:18:21,010
about dual relationships between
supervisory relationships
328
00:18:21,010 --> 00:18:22,135
and personal relationships.
329
00:18:22,830 --> 00:18:25,160
But anyway, let's
say if we can say
330
00:18:25,160 --> 00:18:29,980
that profs should not teach
anyone one that they're
331
00:18:29,980 --> 00:18:31,100
advising.
332
00:18:31,100 --> 00:18:35,370
Well, if we were saying that in
logical notation, what we would
333
00:18:35,370 --> 00:18:38,700
say is that for every
professor and subject,
334
00:18:38,700 --> 00:18:43,310
it's not the case
that the professor has
335
00:18:43,310 --> 00:18:46,070
an advisee in subject
j and the professor
336
00:18:46,070 --> 00:18:48,124
is teaching subject j.
337
00:18:48,124 --> 00:18:49,790
So that's how you
would say it in logic,
338
00:18:49,790 --> 00:18:51,790
but there's a very
slick way to say it
339
00:18:51,790 --> 00:18:54,620
without all the formulas
and the quantifiers.
340
00:18:54,620 --> 00:18:59,880
I could just say that T, the
relationship of his teaching,
341
00:18:59,880 --> 00:19:02,980
intersected with
the relationship
342
00:19:02,980 --> 00:19:07,150
of has an advisee in
the subject is empty.
343
00:19:07,150 --> 00:19:14,210
There is no pair of professor
and subject that is in both T
344
00:19:14,210 --> 00:19:17,150
and in R of V.
345
00:19:17,150 --> 00:19:20,740
And this bottom
expression here gives you
346
00:19:20,740 --> 00:19:22,700
a sense of the
concise way that you
347
00:19:22,700 --> 00:19:26,350
can express queries and
assertions about the database
348
00:19:26,350 --> 00:19:30,760
using a combination of
relational operators
349
00:19:30,760 --> 00:19:32,870
and set operators.
350
00:19:32,870 --> 00:19:34,860
Another way to say it
by the way-- there's
351
00:19:34,860 --> 00:19:37,490
a general set theoretic
fact-- is the way
352
00:19:37,490 --> 00:19:41,870
to say that T and R of
V intersected is empty
353
00:19:41,870 --> 00:19:45,630
is to say that the set T and the
set R of V, whatever they are,
354
00:19:45,630 --> 00:19:48,460
have no points in common.
355
00:19:48,460 --> 00:19:50,780
An equivalent way to
say that is that one set
356
00:19:50,780 --> 00:19:53,490
is contained in the
complement of the other set.
357
00:19:53,490 --> 00:19:57,060
So I could equally well have
said this as R composed with V
358
00:19:57,060 --> 00:20:02,310
is a subset of not T.
359
00:20:02,310 --> 00:20:04,330
Well, let's step back
now and summarize
360
00:20:04,330 --> 00:20:07,370
what we've done by example
and say a little bit
361
00:20:07,370 --> 00:20:09,170
about how it works in general.
362
00:20:09,170 --> 00:20:12,020
So as I said, a binary
relation-- and we'll
363
00:20:12,020 --> 00:20:14,470
start to be slightly more
formal now-- a binary relation
364
00:20:14,470 --> 00:20:19,250
R from a set A to a set B
associates elements of A
365
00:20:19,250 --> 00:20:21,330
with elements of B.
366
00:20:21,330 --> 00:20:24,570
And there's a picture
of a general set
367
00:20:24,570 --> 00:20:27,880
A called the domain
and a general set B
368
00:20:27,880 --> 00:20:29,590
called the codomain.
369
00:20:29,590 --> 00:20:33,090
And R is given by those arrows.
370
00:20:33,090 --> 00:20:36,150
Well, what exactly are arrows?
371
00:20:36,150 --> 00:20:38,840
Well, if you're going
to formalize arrows,
372
00:20:38,840 --> 00:20:41,610
the set of them is what's
called the graph of R.
373
00:20:41,610 --> 00:20:45,710
So technically, a relation
really has three parts.
374
00:20:45,710 --> 00:20:48,860
It's not to be identified
with just its arrows.
375
00:20:48,860 --> 00:20:51,620
A relation has a
domain and codomain
376
00:20:51,620 --> 00:20:54,194
and some bunch of arrows
going from the domain
377
00:20:54,194 --> 00:20:54,860
to the codomain.
378
00:20:57,940 --> 00:21:01,400
The arrows can be
formalized by saying all
379
00:21:01,400 --> 00:21:04,430
that matters about an arrow is
where it begins where it ends
380
00:21:04,430 --> 00:21:07,430
because it's just
designed to reflect
381
00:21:07,430 --> 00:21:10,520
an association between
an element of the domain
382
00:21:10,520 --> 00:21:12,370
and an element of the codomain.
383
00:21:12,370 --> 00:21:15,840
So technically, the arrows
are just ordered pairs.
384
00:21:15,840 --> 00:21:18,170
And in this case, there are
three arrows-- one from A
385
00:21:18,170 --> 00:21:19,230
to b 2.
386
00:21:19,230 --> 00:21:23,770
And so you see at the bottom of
the slide an ordered pair a 1,
387
00:21:23,770 --> 00:21:25,290
b 2.
388
00:21:25,290 --> 00:21:27,700
Another arrow goes
for a 1 to b 4.
389
00:21:27,700 --> 00:21:29,445
So you see the
ordered pair a 1, b 4.
390
00:21:29,445 --> 00:21:32,680
And the final arrow is a 3, b 4.
391
00:21:32,680 --> 00:21:34,300
And you see that pair.
392
00:21:34,300 --> 00:21:36,420
So all the language
about arrows is really
393
00:21:36,420 --> 00:21:39,140
talking about ordered pairs.
394
00:21:39,140 --> 00:21:43,220
It's just that the
geometric image
395
00:21:43,220 --> 00:21:45,750
of these diagrams
and their arrows
396
00:21:45,750 --> 00:21:50,170
makes a lot of
properties much clearer.
397
00:21:50,170 --> 00:21:55,010
So the range of R is
an important concept
398
00:21:55,010 --> 00:21:57,620
that comes up regularly
and tends to be
399
00:21:57,620 --> 00:21:59,330
a little confusing for people.
400
00:21:59,330 --> 00:22:01,040
The range of R is
simply the elements
401
00:22:01,040 --> 00:22:04,630
with arrows coming
in from R. It's
402
00:22:04,630 --> 00:22:09,440
all of the elements that
are hit by an arrow that
403
00:22:09,440 --> 00:22:11,980
starts in the domain.
404
00:22:11,980 --> 00:22:17,400
So it's really R of the
domain is the range of R.
405
00:22:17,400 --> 00:22:21,040
Now, notice that
this is typically not
406
00:22:21,040 --> 00:22:23,790
equal to the whole codomain.
407
00:22:23,790 --> 00:22:25,310
Let's look at this example.
408
00:22:25,310 --> 00:22:28,430
Here, the range
of R-- the points
409
00:22:28,430 --> 00:22:32,740
that are hit by elements of
A under R, namely just b 2
410
00:22:32,740 --> 00:22:34,460
and b 4.
411
00:22:34,460 --> 00:22:38,506
The codomain has elements b
1 and b 3 that are missing
412
00:22:38,506 --> 00:22:39,755
and that are not in the range.
413
00:22:42,310 --> 00:22:45,050
Well, as I said, functions are
a special case of relations.
414
00:22:45,050 --> 00:22:47,130
So let's just look at that.
415
00:22:47,130 --> 00:22:51,240
A function, F, from
a set A to a set B
416
00:22:51,240 --> 00:22:54,500
is a relation which
associates with each element
417
00:22:54,500 --> 00:22:57,255
in the domain-- each element
little a and the domain capital
418
00:22:57,255 --> 00:23:04,350
A-- with at most one
element of the codomain B.
419
00:23:04,350 --> 00:23:07,730
So this one element, if it
exists, is called F of a.
420
00:23:07,730 --> 00:23:13,190
It's the image of the element
a under the relation F,
421
00:23:13,190 --> 00:23:17,120
but what's special about it is
that F of a contains at most
422
00:23:17,120 --> 00:23:18,390
one element.
423
00:23:18,390 --> 00:23:21,110
So let's just look
at an example again.
424
00:23:21,110 --> 00:23:24,080
A way to say that a
relation is a function
425
00:23:24,080 --> 00:23:29,320
is to look at all of the points
on the left in the domain
426
00:23:29,320 --> 00:23:33,530
and make sure that none of them
have more than one arrow coming
427
00:23:33,530 --> 00:23:34,180
out.
428
00:23:34,180 --> 00:23:37,810
Well, in this picture, there are
a couple of violations of that.
429
00:23:37,810 --> 00:23:40,060
There are a couple
points on the left in A
430
00:23:40,060 --> 00:23:42,220
that have more than
one arrow coming out.
431
00:23:42,220 --> 00:23:45,070
[? There's ?] our two bad edges.
432
00:23:45,070 --> 00:23:48,960
But if I erase those, now
I'm left with a function.
433
00:23:48,960 --> 00:23:52,160
And sure enough, there's at
most one arrow coming out
434
00:23:52,160 --> 00:23:55,320
of each of the points on the
left in A. Some of the points
435
00:23:55,320 --> 00:23:56,510
have no arrows coming out.
436
00:23:56,510 --> 00:23:57,870
That's fine.
437
00:23:57,870 --> 00:24:02,200
And so for those green
points with an arrow out,
438
00:24:02,200 --> 00:24:07,460
there's a unique F of the
green point equal to a magenta
439
00:24:07,460 --> 00:24:11,980
point in B that's uniquely
determined by the functional
440
00:24:11,980 --> 00:24:16,010
relation F, which may
not be defined for all
441
00:24:16,010 --> 00:24:18,791
of the green points if they
don't have any arrow coming out
442
00:24:18,791 --> 00:24:19,290
of them.
443
00:24:19,290 --> 00:24:23,605
So function means less than or
equal to 1 arrow coming out.
444
00:24:26,210 --> 00:24:29,020
So if we set this
formally without talking
445
00:24:29,020 --> 00:24:31,620
about the arrows,
one way is simply
446
00:24:31,620 --> 00:24:36,980
to say that a
relation is a function
447
00:24:36,980 --> 00:24:40,460
if the size of F of little a is
less than or equal to 1 for all
448
00:24:40,460 --> 00:24:44,470
of the domain elements A.
449
00:24:44,470 --> 00:24:48,150
And a more elementary
way to say it
450
00:24:48,150 --> 00:24:51,130
using just the language
of relations and equality
451
00:24:51,130 --> 00:24:53,940
and Boolean
connectives is to say
452
00:24:53,940 --> 00:24:57,500
that if a is connected
to two things
453
00:24:57,500 --> 00:25:03,110
by F-- if aFb AND aFb
prime-- then in fact
454
00:25:03,110 --> 00:25:06,690
b is equal to b prime.
455
00:25:06,690 --> 00:25:10,680
And that wraps up functions.