MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 4
is said to be a linear order (in the strict sense) if it is:
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 9
be the set of all chains of a given poset
. Notice that the elements of
are chains in ,
Problem 1 Let
but we can also speak of chains in
(in the sense of Zorns Lemma). To avoid con
MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 2
and
Problem 2.12 Both
Venn diagram:
correspond to the same shaded region in the following
A
C
B
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Problem 2.16 Notice that for any sets and , one has
MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 8
be the following ordering on the integers :
Problem 7.11 (a) Let
Clearly, this is a linear ordering. To see that it is a well-ordering, consider any non-empty subset of . Case
MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 5
is a transitive set. Then
by Theorem 4E. This shows that
Problem 4.2 Suppose
is transitive.
Problem 4.3
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; it sufces to show that
, thus
, as desired.
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 1
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Problem 1.4 Assume that
follows that
and
. It
Problem 1.5 We have:
1.
,
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 6 (Revised)
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Problem 4.19 We want to prove the claim: for all natural numbers and , if
, then there exists natural
numbers
such that
and
. Fix some
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Well-Orders and Ordinals
1 Transnite Induction and Recursion
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Theorem 1 (Transnite Induction Principle). Let
is a subset such that for all
,
Suppose
0
, let us write
is well-ordered if
MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 7
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to Problem Set 10
(
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(
(
be any sets such that for all ,
), this implies
, hence, since
. This holds for no . Moreover, if
is
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MATH 582, INTRODUCTION TO SET THEORY, WINTER 1999
Answers to the First Midterm
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