Chapter 5 Set Theory

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Chapter 5 Set Theory
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5.1 Basic Definitions
Introduction Roughly speaking, a set is a
collection of objects, but in set theory, the
words set and element are intentionally left
as undefined. There is also another undefined
relation ?, called the membership relation. If
S is a set and a is an element of S, then we
write a ?S, and we can say that a belongs to S.
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5.1 Basic Definitions
Introduction The notation If a set M has
only a finite number of elements say, 3, 7, and
11, then we can write
M 3, 7, 11 (the order in which they appear is
unimportant.) A set can also be specified by a
defining property, for instance
S x ? ? -2 lt x lt 5 (this is
almost always the way to define an infinite set).
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5.1 Basic Definitions
Introduction Some basic axioms Axiom of
existence there is a set. Axiom of extension
For any two sets A and B,
In other words, two sets are equal if and only if
they have the same elements. For example, 1, 2,
3 3, 1, 2 1, 2, 3, 2
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• Axioms of Zermelo-Fraenkel
• Axiom of Extensionality
• Axiom of Pairing
• Axiom Schema of Separation.If ? is a property
  with parameter p, then for any set X, there
  exists a set Y u ? X ?(u, p)
• Axiom of UnionFor any set X, there is a set Y
  that is the union of all elements in X.
• Axiom of Power SetFor any set X, there exists a
  set Y that contains all subsets of X.
• Axiom of InfinityThere is an infinite set.
• Axiom Schema of ReplacementIf F is a function,
  then for any set X, there is a set Y F(x) x
  ?X
• Axiom of Regularity Every non-empty set has an
  ?-minimal element.
• Axiom of ChoiceThe Cartesian product of any
  non-empty family of non-empty sets is non-empty.

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5.1 Basic Definitions
Subsets Given two sets A and B, we say that A
is a subset of B, denoted by
if
In other words, A is a subset of B if all
elements in A are also in B.
Lemma Two sets A and B are equal if A is a
subset of B and B is a subset of A. i.e.
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5.1 Basic Definitions
Proper Subsets Given two sets A and B, we say
that A is a proper subset of B, denoted by
if
In other words, A is a proper subset of B if all
elements in A are also in B but A is smaller
than B.
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5.1 Basic Definitions
Operations of Sets Let A and B be two subsets
of a larger set U, we can define the
following, 1. Union of A and B,
2. Intersection of A and B,
3. Difference of B minus A,
4. Complement of A,
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5.1 Basic Definitions
Ordered pairs For any two elements a and b (not
necessarily distinct), we define
(a, b) a, a, b which
is called the ordered pair of a and b. It is not
hard to see that (a, b) (c, d)
if and only if (a c and b d) In other
words, the order of the elements is
important. We can similarly define ordered
triples and ordered n-tuples
(x1, x2, x3, , xn)
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5.1 Basic Definitions
Cartesian Products For any two sets A and B, the
Cartesian Product of A and B, denoted by AB
(read A cross B), is the set of all ordered pairs
of the form (a, b) where a?A and b?B.
Given sets A1, A2 , , An we can define the
Cartesian product
A1 A2 An as the set of all
ordered n-tuples.
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5.2 Properties of Sets
Theorem 5.2.1 1. Inclusion of Intersection For
all sets A and B,
2. Inclusion in Union For all sets A and B,
3. Transitive property of Subsets For all sets
A, B, and C,
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5.2 Properties of Sets
Theorem 5.2.2 Set Identities Let all sets
referred to below be subsets of a set U. 1.
Commutative Laws For all sets A and B
2. Associative Laws For all sets A, B, and C,
3. Distributive Laws For all sets A, B, and C,
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5.2 Properties of Sets
Theorem 5.2.2 Set Identities Let all sets
referred to below be subsets of a set U. 4.
Intersection with U For all sets A
5. Double complement Law For all sets A
6. Idempotent Laws For all sets A,
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5.2 Properties of Sets
Theorem 5.2.2 Set Identities Let all sets
referred to below be subsets of a set U. 7.
DeMorgans Law For all sets A and B,
8. Union with U For all sets A
9. Absorption Laws For all sets A and B,
10. Alternate Representation for set difference
For all sets A and B,
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5.3 The empty set, Partition, and Power sets
Definition A set with no element in it is called
an empty set.
Theorem If Ø is an empty set, then it is a subset
of any set.
Corollary There is only one empty set.
Notation Since there is only one empty set, we
use the symbol Ø to denote this unique empty set,
and we will call it the empty set.
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5.3 The empty set, Partition, and Power sets
Properties of the empty set Let all sets
referred to below be subsets of a set U. 1. Union
with Ø
2. Intersection and union with the complement
3. Intersection with Ø
4. Complement of U and Ø
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5.3 The empty set, Partition, and Power sets
Definition Two sets A and B are said to be
disjoint if their intersection is empty.
Proposition Given any two sets A and B,
Definition A collection of sets A1, A2 , An ,
are said to be mutually disjoint (or pairwise
disjoint) if
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5.3 The empty set, Partition, and Power sets
Definition A collection of non-empty sets A1,
A2 , An is a partition of a set A if
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5.3 The empty set, Partition, and Power sets
Definition Given a set A, the power set of A,
denoted , is the set of all subsets
of A.
(the existence of such a power set is an axiom,
and cannot be proved from other previous axioms)
Theorem For all sets A and B,
Theorem If A is a set with n elements, then
has 2n elements.