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Chapter 3 Set Theory
3.1 Sets and Subsets
A well-defined collection of objects
(the set of outstanding people, outstanding is very subjective)
有限集合 , 無限集合 , 一個集合的基數 , 子集合
A={1,3,5,7,9}B={x|x is odd}C={1,3,5,7,9,...}A 的基數 (|A|=5)A 為 B 的真子集 .C 為 B 的子集合 .
1 1 1 A B C, ,
A B
C B
3
Chapter 3 Set Theory
3.1 Sets and Subsets
相等的 C D C D D C ( ) ( )
子集合 A B x x A x B [ ]
A B x x A x B
x x A x B
x x A x B
[ ]
[ ( ) )]
[ ]
C D C D D C
C D D C
( )
4
Chapter 3 Set Theory
3.1 Sets and Subsets
零集或空集合 : {},
宇集 : U
A 的冪集合 :A 的所有子集合所成的集合
A={1,2}, P(A)={, {1}, {2}, {1,2}}
If |A|=n, then |P(A)|=2n.
5
If |A|=n, then |P(A)|=2n.
Chapter 3 Set Theory
3.1 Sets and Subsets
對任一有限集合 A , |A|=n0, 共有 C(n,k) 個大小為 K 的子集合
依據子集合的元素 K ,計數 A 的子集合,我們得合
0for ,2210
n
n
nnnn n
6
Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.10 Number of nonreturn-Manhattan paths betweentwo points with integer coordinated
由 (2,1) 到 (7,4): 3 向上 , 5 向右
8!/(5!3!)=56R,U,R,R,U,R,R,Upermutation
8 個 路徑 , 選出3個路徑向上{1,2,3,4,5,6,7,8}, 一個三元件子集合表示一個方法 ,例如 , {1,3,7} 表示路 徑 1, 3, and 7 為向上 .許多三元件子集合 =C(8,3)=8!/(5!3!)=56
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Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.11 一個正整數的許多合成4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1
4 有 8 個合成 .(4 有 5 個分割 .).
Consider 4=1+1+1+1
第一個 加號
第二個 加號
第三個 加號
The uses or not-uses ofthese signs determinecompositions.
合成 = 子集合 {1,2,3}=8
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Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.12 對整數 n , r 及n r 1
prove n
r
n
r
n
r
1
1combinatorially.
Let A x a a an{ , , , , }1 2
考慮含 r 個元素的 A 的所有子集合:n
r
n
r
n
r
1
1
不包含 x所有可能的 包含 x
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Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.14 巴斯卡三角形0
01
0
1
1
2
1
2
2
2
0
3
2
3
1
3
3
3
0
4
1
4
2
4
3
4
4
4
0
binomialcoefficients
10
Chapter 3 Set Theory
3.1 Sets and Subsets
common notations
(a) Z= 所有整數的集合 ={0,1,-1,2,-1,3,-3,...}(b) N= 所有非負整數或自然數所成的集合(c) Z+= 所有正整數所成的集合(d) Q= 所有有理數所成的集合 ={a/b| a,b is integer, b not zero}(e) Q+= 所有正有理數所成的集合(f) Q*= 所有非零實數所成的集合(g) R= 所有實數所成的集合(h) R+= 所有正實數所成的集合(i) R*= 所有非零實數所成的集合(j) C= 所有複數所成的集合
11
Chapter 3 Set Theory
3.1 Sets and Subsets
common notations
(k) C*= 所有非零複數所成的集合(l) For any n in Z+, Zn={0,1,2,3,...,n-1}(m) 對每個實數 a,b with a<b,
[ , ] { | }a b x R a x b ( , ) { | }a b x R a x b
[ , ) { | }a b x R a x b
( , ] { | }a b x R a x b
closed interval
open interval
half-open interval
13
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Def. 3.5 For A,BU
a) A B x x A x B
A B x x A x B
A B x x A B x A B
{ | }
{ | }
{ | }b)c)
聯集交集對稱差集
Def.3.6 互斥 A B
Def 3.7 餘集 A U A x x U x A { | }
Def 3.8 A 在 B 的(相對)餘集B A x x B x A { | }
14
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
對任意宇集 U 及任意集合 A,B in U ,下面敘述為等價的︰
A B
A B B
A B A
B A
a)
b)c)
d)
reasoning process
(a) (b), (b) (c),
(c) (d), and (d) (a)
15
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
集合論定律
)()()(
Laws )()()( (5)
)()(
Laws )()( (4)
Laws (3)
Laws ' (2)
of Law )1(
CABACBA
veDistributiCABACBA
CBACBA
eAssociativCBACBA
ABBA
eCommutativABBA
BABA
sDemorganBABA
ComplementDoubleAA
16
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
集合論定律
A)BA(A
Laws Absorption A)BA(A (10)
Laws Domination =A ,UUA (9)
Laws Inverse AA ,UAA (8)
Laws Identity AUA ,AA (7)
Laws Idempotent AAA ,AAA (6)
17
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
s 對偶 s (sd)
U
U
對偶原理。令 S 表一個處理二個集合表示式相等的定理,則Sd , S 的對偶,亦為一個定理
18
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Ex. 3.17 What is the dual of A B ?
Since A B A B B A B
A B B A B B B A
.
.
The dual of is the dual of
, which is That is, .
U
AA A B
A B范恩圖
19
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set TheoryEx. 3.19. Negate
B
A B
A B x x A x B A
A B A B A B
.
{ | }
Ex. 3.20 Negate A B
A B x x A B x A B
A B A B A B A B
A B A B A B A B A B
A B A B A B A A B B
B A A B A B A B
A B A B
.
{ | }
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) [( ) ] [( ) ]
( ) ( ) ( ) ( )
20
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Def 3.10.
i I
i i
i Ii i
A x x A i I
A x x A i I
{ | }
{ | }
for at least one , and
for every
I: 指標集
Theorem 3.6 一般化的狄摩根定律
i Ii
i Ii
i Ii
i Ii
A A
A A
22
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
在一個50位大一新生的班級上,有30位學習 C++ ,25位學習 JAVA ,且有10位二種語言都學習,試問有多少位大一新生學習 C++ 或 JAVA ?
U A B
10 1520
5| | | | | | | |A B A B A B
23
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
對100個此類門樣本set A: with D1
set B: with D2
set C: with D3
Ex 3.26. :AND 門有任何或所有下面的缺陷D1: 輸入 I1 卡住在0。 D2: 輸入 I 2卡住在 0D3: 輸入 O 卡住在 1
with |A|=23, |B|=26, |C|=30,| | , | | , | | ,| |A B A C B CA B C
7 8 103, 有多少個門至少有一個缺陷 ?
A
B
C
11 43
57
12
15
43
Ans:57
| | | | | | | | | || | | | | |A B C A B C A B
A C B C A B C
24
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
Ex 3.27 有三種遊戲,有多少種方法一位學生可每天玩一種遊戲,使得五天內,這三種遊戲他每種至少玩一次?
set A: 不玩第一種遊戲set B: 不玩第二種遊戲set C: 不玩第三種遊戲| | | | | |
| | | | | || |
| |
A B C
A B B C C AA B C
A B C
Ans
2
10
3 2 3 1 0 93
3 93 150
5
5
5 5
5
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