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Equivalence relations
are another kind

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of binary relation
on a set which

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play a crucial role in
mathematics and in computer

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science in particular.

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And they can also be explained
both in terms of digraphs

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and in terms of axioms.

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So let's begin with
a digraph explanation

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of an equivalence relation.

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And the kind of relation
that's an equivalence relation

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is the relation of there being
a walk in both directions

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between two vertices.

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So if there's a walk between
vertex u and vertex v

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and conversely there's a walk
from vertex v back to vertex u,

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then u and v are said to
be strongly connected,

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and strongly connected
is going to be

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an example of an
equivalence relation.

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So in terms of
the walk relation,

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including 0-length walks, the
relation we're talking about is

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u G* v and v G* u.

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Now, as a property of
relations, this has a name.

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It's called symmetry.

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So a relation R on a set A is
symmetric if and only if a R

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b implies b R a,
and the first remark

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is that the strongly connected
relation is symmetric.

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An equivalence relation is
a symmetric relation that

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is transitive and reflexive.

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And again, we have immediately
that the walk relation--

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the mutual walk relation,
the two-way walk relation

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or strongly connected
relation in a digraph

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is an equivalence relation.

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Because clearly if there's
two way paths between u and v

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and between and v and w, then
there's one between u and w

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by going for u to
v to w and back.

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Likewise, there
is a length 0 walk

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from any element to itself.

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And by definition, strong
connectedness is symmetric.

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So the strong connectedness
relation in any digraph

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is an equivalence relation.

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And the theorem is, conversely,
that any equivalence relation,

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anything that's an
equivalence relation,

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is the strongly connected
relation of some digraph.

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The proof is trivial.

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It's the strongly connected
relation of itself.

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OK.

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Some examples of
equivalence relations

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to see why they're so basic is
that the most fundamental one

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is equality.

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Obviously, equality is symmetric
and reflexive and transitive,

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and so it's an
equivalence relation.

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Another one that we've
seen is congruence mod n,

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which you could also check is
symmetric and transitive and

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reflexive.

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And finally, another
relation would

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be that two sets
are the same size,

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providing they're finite sets.

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And another example would
be a bunch of objects

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having the same color.

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Two objects have the same color
is a relation among objects

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that have color that is
symmetric and transitive and

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reflexive, so it's an
equivalence relation.

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Let's illustrate
some of these axioms

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that we have in terms of graphs.

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It can be helpful
to remember them.

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So reflexive means that
when you look at a digraph,

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it's reflexive when
there's a little self loop

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from every vertex to itself.

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So there's a length 1 path or
an edge from vertex to itself

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in reflexive graphs.

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Transitive means that whenever
you have two edges connecting

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one vertex to another,
there's a path

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of length 2 from
one place to another

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that in fact is an edge from
that place to its target.

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And of course as
we said, once there

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is an edge wherever
there's a path of length 2,

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it follows by
induction that there's

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an edge wherever there's
a path of any length,

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and that's what
transitive means.

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Asymmetric means
that whenever you

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have an edge from one vertex to
another there is no edge back.

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So in particular,
if I have an edge

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from this vertex to
that vertex in blue,

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there is no edge that goes
back in the other direction.

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Nor is there ever a self
loop in an asymmetric graph.

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And finally, in a
symmetric graph,

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wherever there's
an edge, there's

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an edge that goes
back the other way.

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So that can help
you maybe remember

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what these properties mean.

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Now again,
equivalence relations,

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besides being
represented in terms

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of the strongly connected
relation of a digraph,

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can be represented in two
other very natural ways that

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really explains
where they come from

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and what their properties are.

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So whenever you have a
total function f on a set A,

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it defines an equivalence
relation on the set A. Namely,

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if f is a total function
from domain A to codomain B,

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then we can define a relation
we can call equivalence sub f

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on the set A by the rule that
two elements are equivalents

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of f if and only if they
have the same image under f--

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they hit the same thing.

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That is, A is equivalent
sub f to A prime if

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and only if f of a is
equal to f of a prime.

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And again, equivalence
sub f immediately

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inherits the properties
of equality, which makes

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it an equivalence relation.

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And the theorem that we have is
that every relation R on a set

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A is an equivalence relation
if and only if it in fact

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is equal to equivalence
sub f for some function f.

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Let's illustrate that.

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We already remembered
that congruence mod n

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can be understood as
equivalence sub f,

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where the mapping is just
map things to remainders.

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Two numbers are congruent
mod n if and only

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if they have the same
remainder on division by n.

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So map a number a to f of
k, equal its remainder,

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and we have found
the equivalence sub f

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representation of
congruence, which

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is another way to
verify that congruence

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is an equivalence relation.

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Finally, whenever you
have a partition of a set,

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you can define an
equivalence relation.

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So a partition of a
set cuts up the set A

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into a bunch of blocks
which are nonempty,

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and every element is a
member of some block,

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and the blocks don't overlap.

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So in fact, every element is
a member of a unique block.

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And that enables me to define
an equivalence relation on A

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by the property that two
elements are in the same block.

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In fact, that's the proof of the
previous representation theorem

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in terms of a function that you
can map an element to the block

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that it's in, in order
to see that the block

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representation and
the equivalence sub f

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representation are the same.

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The proof in the
other direction,

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that every equivalence relation
can be represented in this way,

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is an exercise in
axiomatic reasoning,

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and elementary one that we're
going to leave to a problem

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and not do in this presentation.

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So the theorem finally is that,
again, a relation R on a set

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is an equivalence
relation if and only

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if it is in fact the
being in the same block

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relation for some partition.

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And that is the story and
multiple ways of understanding

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what equivalence relations are.