The document defines basic set theory concepts and operations such as sets, subsets, union, intersection, complement and Venn diagrams. It provides examples to illustrate set equality, subset, union, intersection and complement relationships. Formulas for calculating set sizes and operations on multiple sets are also presented.

1. 1

2.  set
A, B

 a A “a A”
a A “a A”
 A = {a1, a2, …, an} “A a1, …, an”

2

3. 
:
 a, b, c
{a, b, c} = {a, c, b} = {b, a, c} =
{b, c, a} = {c, a, b} = {c, b, a}

 a=b, {a, b, c} = {a, c} = {b, c} =
{a, a, b, a, b, c, c, c, c}
 !
3

4.  A B
:
4
• A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = B
• A = { , , },
B = { , , , } : A B
• A = { , , },
B = { , , , } :
A = B
• A = {1, 2, 3, 4},
B = {x | x x>0 x<5
}:
A = B

5.  A = “ ”
 A = {z} : z A, z {z}
 A = {{b, c}, {c, x, d}}
 A = {{x, y}} : {x, y} A, {x, y} {{x,
y}}
 A = {x | P(x)} “ x P(x)
”
P(x) membership function)
A
x (P(x) x A)
 A = {x | x N x > 7} = {8, 9, 10, …}
“ set builder5

6. 
infinite
 :
 Q = {a/b | a Z b Z+}
N = {0, 1, 2, …}
Z = {…, -2, -1, 0, 1, 2, …}
R = {-0.15, 3.67, 30,
74.18284719818125…}

ℕ, ℤ, ℝ
 6

7. “ ”:
 Natural numbers N = {0, 1, 2,
3, …}
 Integers Z = {…, -2, -1, 0, 1, 2, …}
 Positive Integers Z+ = {1,
2, 3, 4, …}
 Real Numbers R = {47.3, -12, , …}
 Rational Numbers Q = {1.5, 2.6,
-3.8, 15, …}
7

8. 

U

8
For 2 sets For 3 sets

9. 9

10. 10

11. 11

12.  (“null”, “ ”)
 = {} = {x|False}
 ,
x: x
12

13.  S T (“S T”)
S T
 S T x (x S x T)
 S
 S S
 S T (“S T”) T S
 : S=T S T S T
 (S T), x(x S x T)
13
TS /

14.  S T (“S T”) S T
 : A = {1, 2, 3}, B = {2, 3, 1},
C = {3}
 B = A, C A, C B
 : U = 1, 2, 3, , 11, 12
T = 1, 2, 3, 6
 T U T U
14
ST /

15. Example: A B, A B
 A={x|x 42 ≤ x ≤ 51}
 B={x|x=4k+3 k N}
 x A x=43 x=47
43=4(10)+3 47=4(11)+3
 A B
A B
#
 B A
15

16. Example: A B B A A
B
 A={3k+1 | k N} B={4k+1 | k N}

A={1,4,7,…}
B={1,5,9,…}
 4 A 4 B A B
5 B 5 A B A
#
16

17. Example: A = B A B
 A={x|x 12 ≤ x ≤ 18}
 B={x|x=4k+1 k {3,4}}
 A B x A x=13
x=17 13=4(3)+1
17=4(4)+1 x B A
B
 B A x B
x=4(3)+1 x=4(4)+1 x=13 17
17

18.  |S| ( “ S”)
S
 , | |=0, |{1,2,3}| = 3, |{a,b}| = 2,
|{{1,2,3},{4,5}}| = ____
 |S| N, S finite
S infinite
18
D = { x N | x 7000 } |D| = 7001
E = { x N | x 7000 } E

19.  P(S) S
S P(S) :≡ {x | x S}
 P({a,b}) = { , {a}, {b}, {a,b}}
 A = , P(A) = { }, : |A| = 0, |P(A)| = 1
 S , |P(S)| = 2|S|
 S:|P(S)|>|S|, |P(N)| > |N|
19

20. 
 n N, n n
(a1, a2, …, an)
a1
 (1, 2) (2, 1) (2, 1, 1)
 n (a1, a2, a3, …, an)
(b1, b2, b3, …, bn)
,
20

21.  A, B, Cartesian
product
A B : {(a, b) | a A b B }.
 A = , A =
 : A = {x, y}, B = {a, b, c}
A B = {(x, a), (x, b), (x, c), (y, a), (y, b), (y,
c)}
 A B :
A B A B B A
 A, B , |A B|=|A||B|
 A n
A1 A2 … An...
21

22.  A, B, nion) A B
A, B (
)
 , A,B: A B = {x | x A x B}
 A B A
B
A, B: (A B A) (A B B)
22
•{a,b,c} {2,3} =__________
•{2,3,5} {3,5,7} =___________
{a,b,c,2,3}
{2,3,5,3,5,7}
= {2,3,5,7}

23.  union: A B
n-ary
union:
= A1 A2 … An

:
 I Ai
i I union
1i
iA
Oddi
iA
23

n
i
iA
1
Ii
iA

24.  A, B, intersection A B
A (“ ”)
B
 , A,B: A B={x | x A x B}
 A B A B:
A, B: (A B A) (A B B)
24
•{a,b,c} {2,3} = ___
•{2,4,6} {3,4,5} = ______{4}

25.  intersection: A B
n-
ary intersection :
= A1 A2 … An

:
 I Ai
i I intersection
1i
iA
Oddi
iA
25

n
i
iA
1
Ii
iA

26.  A B ?
|A B| = |A| |B| |A B|
 :
Mailing List) ? E I M,
I = {s | s }
M = {s | s }
 !
|E| = |I M| = |I| |M| |I M|
26

27.  A, B (disjoint)
intersection)
(A B= )
 :
27

28.  A, B, A B,
A B, A
B :
A B : x x A x B
: x x A x B
 : |A-B| = |A| - |A B|
28
Set A Set B
Set
A B
A−B
A
B

29.  {1,2,3,4,5,6} {2,3,5,7,9,11} =
___________
 Z N {… , −1, 0, 1, 2, … } {0, 1, … }
= {x | x }
= {x | x }
= {… , −3, −2, −1}
29
{1,4,6}

30. 30
A B
UA B
A B = { x | x A x B }

31.  universe of discourse
U
 A A U, A complement
,
A U,
= U A
 , U=N,
A
}|{ AxxA
31
A
,...}7,6,4,2,1,0{}5,3{
A
A

32. A B
A B
A
B
A-B
B-A
A B
32
:
U = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
A= 1, 2, 3, 4, 5 ,
B = 4, 5, 6, 7, 8

33.  Identity : A = A = A U
 Domination): A U = U , A =
 Idempotent): A A = A = A A
 Double complement):
 Commutative): A B = B A , A B = B A
 Distributive): A (B C) = (A B) (A C)
 Associative): A (B C)=(A B) C ,
A (B C)=(A B) C
 Absorption): A (A B) = A, A (A B) = A
 Complement): A = U, A =A
A
33
AA)(

34. E1 = E2 ( Es ),
:
1. E1 E2 E2 E1
2. set builder notation
3. membership table
34

35. : A (B C)=(A B) (A C)
 1: A (B C) (A B) (A C)
 x A (B C), x (A B) (A C)
 x A, x B x C
 1: x B x A B, x (A B) (A C)
 2: x C x A C , x (A B) (A C)
 , x (A B) (A C)
 , A (B C) (A B) (A C)
35

36.  2: (A B) (A C) A (B C)
 x (A B) (A C), x A (B C)
 x A B, x A C
 1: x A B x A x B
 2: x A C x A x C
 x A, x B x C x A (B C)
 , (A B) (A C) A (B C)
 A (B C) (A B) (A C)
(A B) (A C) A (B C)
A (B C) = (A B) (A C) #
36

37. :
:
Unions)
(A B ) C = A (B C )
37

38. :
:
Unions)
(A B ) C = A (B C )
: (A B ) C = {x | x A B x C } (
)
38

39. :
:
Unions)
(A B ) C = A (B C )
: (A B ) C = {x | x A B x C } (
)
= {x | (x A x B ) x C } (
)
39

40. :
:
Unions)
(A B ) C = A (B C )
: (A B ) C = {x | x A B x C } (
)
= {x | (x A x B ) x C } (
)
= {x | x A ( x B x C ) } (
)
40

41. :
:
Unions)
(A B ) C = A (B C )
: (A B ) C = {x | x A B x C } (
)
= {x | (x A x B ) x C } (
)
= {x | x A ( x B x C ) } (
)
= {x | x A x B C ) } (
41

42. :
:
Unions)
(A B ) C = A (B C )
: (A B ) C = {x | x A B x C } (
)
= {x | (x A x B ) x C } (
)
= {x | x A ( x B x C ) } (
)
= {x | x A (x B C ) } (42

43.  set
builder notation logical equivalence
 Proof:
 Q.E.D.
BABA
unionofDef.)
complementofDef.)
lawssMorgan'De)
onintersectiofDef.)(
ofDef.))((
complementofDef.
BAxx
BxAxx
BxAxx
BxAxx
BAxx
BAxxBA
43

44. 


 “1” , “0”

44

45. (A B) B = A B
(A B) B = A B #
45
AA BB AA BB ((AA BB)) BB AA BB
0 0 0 0 0
0 1 1 0 0
1 0 1 1 1
1 1 1 0 0

46. (A B) C = (A C) (B C)
46
A B C AA BB ((AA BB)) CC AA CC BB CC ((AA CC)) ((BB CC))
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1

47.  (A B) B = A B Set builder
notation




47