This document provides an outline and definitions for fundamental concepts in set theory and discrete mathematics, including: 1. Definitions of sets, operations on sets like union and intersection, and relations. 2. Functions, relations, and properties like domains, ranges, and composition. 3. Partial orders, trees, groups, and other algebraic structures.

1. A
2008
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2. Georg Cantor:
1845-1918
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3. Outline
1
2
3
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4. a∈A a A
A, B
A = B iff (∀x)(x ∈ A ↔ x ∈ B).1 (1)
∅.
1
iff: if and only if
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5. {1, 2} 1 2
{1, 2} = {2, 1}.
{1, 2, 2} = {1, 1, 1, 2} = {1, 2}.
{dogs}
{}
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6. ( )
{x : P(x)} ( {x|P(x)}) P
{x : x }
{x ∈ A : P(x)} A P x.
{F (x) : x ∈ A} A F
{2x : x ∈ Z}
{F (x) : P(x)} P F
{x 2 : x }
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7. ( )
A, B A B (subset) A
B A B B A.
A⊆B B ⊇ A. B B B A B
A ⊂ B.
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8. A A ⊆ A, ∅ ⊆ A.
R, Q, Z, N
N⊂Z⊂Q⊂R
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9. U U
U U A A( U ) A
A = {x ∈ U : x ∈ A},
U A
U {A : A ⊆ U}, U
P(U) 2U .
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10. A, B A, B
A B A ∪ B;
A B A ∩ B;
A B A − B.
A ∪ B = {x : (x ∈ A) or (x ∈ B)};
A ∩ B = {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B};
A − B = {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B}.
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11. A∪∅=A
A∩U =A
A∪U =U
A∩∅=∅
A∪A=A
A∩A=A
A∪B =B∪A
A∩B =B∩A
A∪A=U
A∩A=∅
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12. (A) = A
A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A∪B =A∩B
A∩B =A∪B
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13. ( )
P(U) F (U ) F
U A, B ∈ F A ∪ B, A ∈ F
F0 = {∅, U}
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14. Outline
1
2
3
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15. a, b (a, b).
(a, b), (c, d) iff a = c, b = d
A,B
A × B = {(a, b) : (a ∈ A) and (b ∈ B)}.
n A1 · · · An
A1 × A2 × · · · × An
n
i=1 Ai .
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16. A, B, C
A × ∅ = ∅ × A = ∅,
A = ∅, B = ∅ A=B A × B = B × A.
A × (B × C) = (A × B) × C;
A × (B ∪ C) = (A × B) ∪ (A × C);
A × (B ∩ C) = (A × B) ∩ (A × C);
(B ∪ C) × A = (B × A) ∪ (C × A);
(B ∩ C) × A = (B × A) ∩ (C × A).
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17. ( )
A, B A B R A×B
A a B b a, b R (a, b) ∈ R.
aRb R(a, b).
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18. A=B A × A,
∅, idA = {(a, a) : a ∈ A}.
A B R R
dom(R) = {x| y (x, y ) ∈ R};
ran(R) = {y | x (x, y ) ∈ R}.
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19. ( )
A B R B C S
R
R ∼ = {(x, y )|(y , x) ∈ R}.
R S
R ◦ S = {(x, z)| y (xRy ) (ySz)}.
R∼ B A R◦S A C
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20. n-
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21. Example (Russell & Novig: AIMA, Chapter 5)
Consider the following binary constraint problem P
V = {WA, SA, NT , Q, NSW , V , T }
U = {red, green, blue}
C: no neighboring regions have the same color
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22. Outline
1
2
3
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23. ( )
A B R A B x ∈A
y ∈B (x, y ) ∈ R R
dom(R) = A,
x ∈ dom(R) y ∈ ran(R)
(x, y ) ∈ R
f , g, h A B f,
f : A → B, (x, y ) ∈ f , f (x) = y .
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24. ( )
f :A→B b∈B a∈A
b = f (a); f a, a ∈ A, f (a) = f (a )
a=a; f f
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25. f :A→B ran(f ) f
ran(f ) = {b} ⊆ B f cb .
A=∅ A B ∅;
f A A a f (a) = a, f
idA .
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26. n
(n )
n ≥ 1, A n f : An → A
n A 0 A
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27. ( )
f : A → B, g : B → C,
g ◦ f : A → C (g ◦ f )(x) = g(f (x)).a
a
f g
f ◦ g, g ◦ f.
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28. f,g g◦f
g◦f g
g◦f f
g◦f f g
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29. U R (reflexive) x ∈U
(x, x) ∈ R R idU ⊆ R.
U R x, y ∈ U
(x, y ) ∈ R (y , x) ∈ R R = R∼.
U R x, y ∈ U,
(x, y ) ∈ R (y , x) ∈ R x =y R ∩ R ∼ ⊆ idU .
U R x, y , z ∈ U,
(x, y ) ∈ R (y , z) ∈ R (x, z) ∈ R, R ◦ R ⊆ R.
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30. ( )
U R R
R U
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31. R U x ∈U [x]R x
[x]R = {y ∈ U : xRy }. U
R
( )
U R U/R = {[x]R : x ∈ U}.
U R
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32. ( )
π U π ⊆ P(U)
π U
π U,
π
π U
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33. U π U Rπ U
a, b Rπ iff a, b π
U R, U/R U
πR ; πR R.
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34. ( )
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35. ( )
P U, U P
U R R P- P R
U
r (R) R r (R) = R ∪ idA .
s(R) R s(R) = R ∪ R ∼ .
t(R) R
∞
t(R) = R ∪ R 2 ∪ R 3 ∪ · · · = Ri .
i=1
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36. X X ,
a a;
if a b and b a then a = b;
if a b and b c then a c.
X
(partially ordered set, or poset)
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37. An example of poset
The Hasse diagram of (℘({x, y , z}), ⊆)2
2
http://en.wikipedia.org/wiki/Hasse_diagram
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38. Total order and well-order
A partial order
is total (or linear) if for any a, b ∈ X , a b or b a
is a well-order if every nonempty subset Y of X has a least
element
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39. Tree
A (rooted) tree is a poset (T , ) such that
T has a unique least element, called the root
the predecessors of every node are well ordered by
A path on a tree T is a maximally linearly ordered subset of T .
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40. Group
A group is a nonempty set G with a binary operation
◦ : G × G → G such that (a ◦ b) ◦ c = a ◦ (b ◦ c) for all
a, b, c ∈ G. An element e in G is called an identity if e ◦ x = x ◦ e
for any x. A semi-group that has an identity is called a monoid.
A semi-group with an identity e is a group if each element x has
a unique inverse y such that x ◦ y = y ◦ x = e.
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