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PROFESSOR: So in
this short segment,

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we'll talk about some
relational properties

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that I call mapping properties.

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They can also be referred to
as archery [? on ?] relations.

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This segment is
mostly vocabulary.

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There are a half
a dozen concepts

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and words that are
standard in the field

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and that one needs to know to be
able to do discrete math and so

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on.

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The applications will come
in the next short segment

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where we start applying
these properties to counting.

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Although, there'll
be a punchline

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about counting at the
end of this segment.

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So let's go back or proceed.

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And remember that
a binary relation

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is a thing with three parts.

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It's got a domain
illustrated as A here,

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a codomain
illustrated as B here,

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and relationship's
an association

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between domain elements and
codomain elements indicated

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by the arrows, the
arrows being called

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the graph of the relation.

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And we already observed one
aspect of archery and arrows

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that the concept
of a function could

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be captured by saying
that there was less than

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or equal to 1 arrow out of
every element in the domain.

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That implied that there
was a unique other end

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of an arrow out
of a domain point

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called the value of that
point under the relation

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which is in fact a function F.

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So F of green
equals magenta where

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there is an arrow out
of a green element.

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But in this picture--
as is typical--

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not every domain element--
not every green dot--

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has an arrow out if it.

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So this would be an illustration
of a partial function

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where F of a green element
isn't always defined

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if there's no arrow out.

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Well, the general idea
of archery relations

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pursues this function
idea that basically we're

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going to classify relations
according to-- first,

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how many arrows come
out of domain elements?

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Really, in three categories.

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The relations where
there's at most

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one arrow out of
every domain element,

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there's exactly one arrow
out of every domain element,

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or there's at least one arrow
out of every domain element.

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And symmetrically, we're going
to classify codomain relations

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with respect to codomain
in the same way-- relations

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where every codomain element
has greater than or equal to 1.

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Arrow in has exactly
one arrow in,

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or at most, one arrow
in is the other part

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of the classification.

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And various combinations
of these things

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have standard name,
which it turns out

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that you'll need to know.

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So we'll lead you through them.

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OK.

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So let's begin with the
idea of a total relation.

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Total relation means there's
at least one arrow out

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every domain element.

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So if you look at this
picture, it's not quite total

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yet because there are
two green domain elements

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with no arrows out of them.

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So I've just
highlighted them in red,

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and we can fix this by
making them disappear.

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Now I'm left with
a total relation.

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Every domain elements has at
least one arrow coming out

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of it.

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So that's what makes it total.

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Another way to say
total is to say

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that if you look at the
inverse image of the codomain,

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it is equal to the domain.

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That means if you take all
the arrows that are coming out

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of the domain and
you turn them around

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and you look at all the things
that have arrowheads into them,

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it's the entire domain.

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So that's what R inverse
of B-- a nice, slick way

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to say it using relational
operators and sets related

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to applying relations.

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So total and function means that
there's exactly one arrow out,

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and that's probably the most
familiar case of functions.

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And lots of fields just assume
that functions are total,

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but the truth is that
there often is not total.

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And people aren't careful about.

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So let's look at a
calculus-like example.

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Here's a function g that
takes a pair of reals

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and returns a real.

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It maps the real plane
into the real line.

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And the definition of it is g
of x, y is 1 over x minus y.

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Now, the domain
of this function g

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is in fact all the
pairs of reals.

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That's what it means to say
that it goes from R cross R--

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shorthand R squared--
to the codomain R.

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The codomain is the
set of all reals.

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But this g is obviously
not total because 1 over 0

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is not defined, which means
that on the 45 degree line,

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g is not defined.

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g of r, r is not defined.

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So g in fact, is
not a total function

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even though it's familiar.

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And you'd not worry about
partial functions normally.

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You wouldn't notice
that this was partial

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because you're not used to
paying attention to that.

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OK.

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Let's look at a
slight variation.

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This is function g 0 that goes
from some unspecified domain.

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I'll specify it in a
minute to the reals.

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It has exactly the same formula
g of x, y is 1 over x minus y.

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But now, I'm going to tell
you that the domain-- instead

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of being all the reals-- is the
reals except for that 45 degree

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line.

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I just want to get
rid of the bad points

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and not worry about them.

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The minute I do that, I
have these two functions

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relations that have the same
graph but different domains.

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And the result is that I've
removed from the domain of g

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all the bad points.

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I'm left with a
total function g 0.

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OK.

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Let's keep going.

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The next concept
is of a surjection,

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and that's a
relation where there

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is at least one arrow into
every point in the codomain.

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There's at least one arrow
into every point in B.

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Well, again this is a picture
where that doesn't quite

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work because there's at least
one bad point there-- there it

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is in red-- that doesn't
have an arrow in.

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So let's fix things again
by making it disappear.

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Now I'm left with a surjective
relation, or a surjection,

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because in fact, everything
in the codomain in B

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has at least one arrow coming.

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Everything's the
endpoint of an arrow.

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So likewise, we can say
in terms of set operations

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that R is a surjection if and
only if the image of the domain

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is the codomain.

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Or still another way
to say it is-- if

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and only if the
range of the function

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is its entire codomain.

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Remember, the
range of the points

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that are hit--
that's R of A-- it's

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not always equal
to the codomain.

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But when it is, that is
what makes it a surjection.

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All right.

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Injections-- another variation
on the theme-- an injection

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is a relation where
there is at most one

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arrow into every
element in the codomain.

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So looking at this picture now,
this is not quite an injection

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because there are at
least two points here

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that have more than one
arrow coming into them.

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That's what keeps it
from being an injection.

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So let's fix that by deleting
a couple of those edges

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that are crowding up
points, and now I'm

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left with a situation where,
in fact, everything in B

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has at most one an
arrow coming in.

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And so I'm showing you a
picture of an injection.

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And the final
concept is when you

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have all the good properties.

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A bijection is when you
have exactly one arrow out

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and exactly one arrow in.

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It's a total function that is
an injection and a surjection

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because it's got greater
than or equal to 1

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and less than or equal
to 1 and equal to 1

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for all of the
domains and codomains.

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Now, there's an obvious thing
though about bijections,

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which we'll wrap
up with, which is

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why they're useful
in counting theory

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because it's clear that since
there's exactly one arrow out

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of every element in A--
the number of arrows

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is the same as the size of A--
and since there's exactly one

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arrow coming into every element
of B, the number of arrows

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is the same as the size of B.

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And guess what.

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That means that where
there's a bijection,

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the sets are of equal size.

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If there's a bijection between
two finite sets A and B,

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that means that
they're the same size.