WEBVTT
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We will now discuss De Morgan's
laws that are some
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very useful relations between
sets and their complements.
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One of the De Morgan's
laws takes this form.
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If we take the intersection of
two sets and then take the
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complement of this intersection,
what we obtain
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is the union of the complements
of the two sets.
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Pictorially, here is
the situation.
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We have our universal set.
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Inside that set, we have a set,
S, which is this one.
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And we have another set,
T, which is this one.
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Let us look at this side.
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The complement of S is this
part of the diagram.
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The complement of T is this
part of the diagram.
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What is left?
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What is left is just this region
here, which is the
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intersection of S with T. So
anything that does not belong
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here belongs to the
intersection.
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This means that the complement
of the intersection is
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everything out there,
which is the set.
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If you're not convinced by this
pictorial proof, let us
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go through an argument that
is a little more formal.
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What does it take for
an element to belong
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to the first set?
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In order to belong to that
set, x belongs to the
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complement of S intersection T.
This is the same as saying
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that x does not belong to
the intersection [of]
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S with T.
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What does that mean?
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Since it is not in the
intersection, this is the same
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as saying that x does not belong
to S or x does not
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belong to T. But this is the
same as saying that x belongs
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to the complement of S or x
belongs to the complement of
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T. And this is equivalent to
saying that x belongs to the
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union of the complement of S
with the complement of T.
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So this establishes this
first De Morgan's law.
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There's another De Morgan's law,
which is obtained from
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this one by a syntactic
substitution.
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We're going to play the
following trick.
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Wherever we see an S, we're
going to replace it by S
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complement.
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And wherever we see an S
complement, we will replace it
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with an S.
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And similarly, whenever we see
a T, we'll replace it by T
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complement.
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And when we see a T complement,
we will replace it
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by T. So doing this syntactic
substitution, what we obtain
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is S complement intersection
with T complement--
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everything gets complemented--
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is the same as S union T.
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Now, let us take complements
of both sides.
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The complement of a complement
is the set itself.
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So we obtain this.
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And now, we take the complement
of the other side,
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which is this one.
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And this is the second
De Morgan's law.
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It tells us that the complement
of a union is the
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same as the intersection
of the complements.
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We derived it from the first De
Morgan's law by a syntactic
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substitution.
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If you're not convinced, it
would be useful for you to go
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through an argument of this kind
to show that something is
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an element of this set if and
only if it is an element of
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that set as well.
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Finally, it turns out that De
Morgan's laws are valid when
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we take unions or
intersections of
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more than two sets.
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There is a more general form.
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And the general form
is as follows--
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an analogy with this one.
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If we have a collection of sets,
Sn, perhaps an infinite
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collection, we take the
intersection of those sets and
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then the complement, what that
is is the union of the
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complements.
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So this is analygous
to this law.
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And this law extends to this
one: if we have the union of
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certain sets and we take the
complement of the union, what
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we obtain is the intersection
of the complements.
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We will have many occasions
to use De Morgan's laws.
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They're actually very useful.
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They allow us, in general, to
go back and forth between
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unions and intersections.