Sets II
Margaret M. Fleck
15 February 2010
This lecture shows how to prove facts about set equality and set inclusion.
(Rosen section 2.2).
1
Announcements
First midterm is a week from Wednesday (i.e. February 24th) 79pm in 141
Wohlers.
Please tell me ASAP about any conflicts or special needs.
For
conflicts, please include a copy of your class schedule.
Quizzes will be handed out in sections today and tomorrow.
Grades
should be on compass this afternoon.
Check out the Exams web page for
information on how to interpret your score.
2
Recap
Last class, we saw a bunch of set theory notation and operations on sets.
Much of this was (more or less) familiar. The least familiar operations were
•
Powerset:
P
(
A
) =
{
all subsets of
A
}
•
Cartesian product:
A
×
B
=
{
(
x,y
)

x
∈
A
and
y
∈
B
}
1
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Remember that (
x,y
) is an ordered pair containing two objects, which
is very different from the twoobject set
{
x,y
}
.
In (
x,y
), the order of the
two elements matters and you can have duplicates e.g. (2
,
2). If you add the
same thing to a set twice, you only get one copy in the set, e.g.
{
2
,
2
}
=
{
2
}
.
3
Set identities
Rosen lists a large number of identities showing when two sequences of set
operations yield the same output sets. For example:
DeMorgan’s Law:
A
∪
B
=
A
∩
B
I won’t go through these in detail because they are largely identical to
the identities you saw for logical operations, if you make the following corre
spondences:
• ∪
is like
∨
• ∩
is like
∧
•
A
is like
¬
P
• ∅
(the empty set) is like
F
•
U
(the universal set) is like
T
The two systems aren’t exactly the same. E.g. set theory doesn’t use a
close analog of the
→
operator. But they are very similar.
You can use these set theory identities to prove new identites via a chain
of equalities.
This is exactly like what you did with logical identities and
there’s no point in walking you through the same exercise twice.
I’m also going to skip showing you set membership tables, which are
in Rosen, because those are just like truth tables and I’m confident you’re
already on top of that idea.
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 Spring '08
 Erickson
 Set Theory, Sets, Basic concepts in set theory

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