Relation and Order

學數學

前言

學 學 數學 學 數學 . 學數學

論 . 學 , .

(Logic), (Set) 數 (Function) . , 學

論. 論 學 數學 .

, ,

. , .

, . , 論

, . , .

v

Chapter 4

Relation and Order

relation. Relation relation

relation. set relation.

relation, equivalence relation. Equivalence relation

, equivalence relation

set . 學 relation .

relation, order. Order ( ),

set .

4.1. Relation

sets X,Y . S X×Y nonempty subset, S relation

from X to Y . , X× X nonempty subset S relation on X.

relation S , x∼ y (x, y) S .

x∼ y x y , relation .

relation , (x, y)∈ S , .

Example 4.1.1. (1) X ={1,0,−1}, Y = {0,1,2}. S ={(x,y) ∈ X ×Y : y = x²+ 1}.

S X Y relation. relation 1∼ 2, 0 ∼ 1 −1 ∼ 2.

(2) X ={1,0,−1}. S ={(x,x^′)∈ X × X : x > x^′}. S X relation.

relation 1∼ 0, 1 ∼ −1 0∼ −1.

Question 4.1. X nonempty set. X relation S. S = X× X x, y∈ X x∼ y.

relation ,

Example 4.1.1(1) X ,Y 數 f (x) = x²+ 1 relation.

relation, , Example 4.1.1(2)

X relation.

55

relation X×Y , S

X ,Y . X ,Y ,

S X×Y . 論 relation .

Example 4.1.1 ( 數, ) S .

.

Example 4.1.2. X relation X “ ”

?

relation X , X power

set P(X) relation. S⊆ P(X) × P(X) A⊆ B

(A, B)∈ S. S ={(A,B) ∈ P(X) × P(X) : A ⊆ B}. relation

A∼ B A⊆ B .

, 論 relation. S relation

on X, , relation.

Reflexive: S x∈ X x∼ x,

(x, x)∈ S, ∀x ∈ X, relation reflexive.

Symmetric: S x, y∈ X x∼ y, y∼ x, (x, y)∈ S ⇒ (y,x) ∈ S, relation symmetric.

Transitive: S x, y, z∈ X x∼ y y∼ z, x∼ z, ((x, y)∈ S) ∧ ((y,z) ∈ S) ⇒ (x,z) ∈ S,

relation transitive.

, .

(1) S reflexive x∈ X (x, x) S. x∈ X

(x, x)∈ S reflexive. (x, y)∈ S x = y. 言 , S

reflexive, X x, (x, x) S ,

(x, y) x̸= y .

(2) S symmetric (x, y)∈ S (y, x) S.

(x, y) (y, x) S symmetric. 言 , S symmetric,

(x, y)∈ S (y, x)̸∈ S . symmetric,

symmetric.

(3) S transitive (x, y) (y, z) S , (x, z) S.

(x, y), (y, z) (x, z) S transitive. (x, y)∈ S

z∈ X (y, z)∈ S. S (x, y) S transitive. 言

4.1. Relation 57

, S transitive, (x, y), (y, z)∈ S (x, z)̸∈ S

. transitive, transitive.

Example 4.1.3. X ={1,2,3}, relation S⊆ X × X reflexive, symmetric transitive.

(1) S ={(1,1),(2,2)}, S reflexive, 3∈ X (3, 3)̸∈ S.

S ={(1,1),(2,2),(3,3),(1,2)}, S reflexive. S (x, y)

x̸= y ( (1, 2)∈ S) reflexive .

(2) S ={(1,1),(2,2)}, S reflexive, S symmetric. (1, 2),

S ={(1,1),(2,2),(1,2)}, (1, 2)∈ S (2, 1)̸∈ S, S symmetric. S (2, 1) ( S ={(1,1),(2,2),(1,2),(2,1)}) symmetric.

(3) S ={(1,1),(2,2),(1,2)}, S reflexive, symmetric, tran- sitive. (2, 3), S ={(1,1),(2,2),(1,2),(2,3)}, (1, 2), (2, 3)∈ S (1, 3)̸∈ S,

S transitive. S (1, 3) ( S ={(1,1),(2,2),(1,2),(2,3),(1,3)}) transitive.

Example reflexive, symmetric transitive .

. relation ,

. relation reflexive symmetric transitive.

, relation , .

relation reflexive symmetric transitive.

論 symmetric transitive reflexive.

Example 4.1.4. X set, S X relation. S symmetric

transitive, S reflexive? 論 ?

x∼ y, S symmetric, y∼ x. , x∼ y y∼ x,

transitive , x∼ x.

論 , S reflexive. , x∈ X,

y∈ X x∼ y, x∼ x. reflexive , x∈ X, x∼ x.

X x , y∈ X x∼ y,

x∼ x . X ={1,2,3} , S ={(1,1),(2,2),(1,2),(2,1)},

S symmetric transitive. 1 (

1∼ 2), 2 1 ( 2∼ 1), S symmetric transitive (1, 1),

(2, 2) S . , 3 , (3, 3) S .

(3, 3)̸∈ S, S symmetric transitive, S reflexive.

Question 4.2. X ={1,2,3} relation on X reflexive symmetric

, transitive . relation on X reflexive

transitive , symmetric .

Example 4.1.2 relation .

Example 4.1.5. X nonempty set. S ={(A,B) ∈ P(X) × P(X) : A ⊆ B}

P(X) relation. S reflexive. A∈ P(X),

A⊆ A ( Proposition 3.1.4(1)), (A, A)∈ S. S reflexive. S

transitive. (A, B)∈ S (B,C)∈ S, A⊆ B B⊆ C, A⊆ C (

Proposition 3.1.4(2)). (A,C)∈ S, S transitive. S symmetric.

, A, B∈ P(X) (A, B)∈ S (B, A)̸∈ S . A = /0 B = X

. /0∈ P(X) X∈ P(X) /0⊆ X, ( /0, X )∈ S. X̸= /0, X * /0, (X , /0)̸∈ S. S symmetric.

Question 4.3. X nonempty set. S ={(A,B) ∈ P(X) × P(X) : A ∪ B = X}

P(X) relation.

(1) S reflexive ? , S reflexive ?

(2) S symmetric ? , S symmetric ?

(3) S transitive ? , S transitive ?

4.2. Equivalence Relation

relation , equivalence relation.

, equivalence classes, .

數學 , .

cd, cd ?

. , 數學, ,

. . ,

, ;

; , .

X , , x, y ,

x∼ y . , 前 : ,

x∈ X x∼ x. relation reflexive. :

, x∼ y y∼ x.

relation symmetric. : ,

, x∼ y y∼ z x∼ z. relation transitive.

, X relation S reflexive, symmetric transitive

, relation ( x∼ y, x, y ),

. reflexive, symmetric

transitive relation . X

relation X× X subset , , ∼ X

relation .

4.2. Equivalence Relation 59

Definition 4.2.1. ∼ X relation, , relation

equivalence relation

Reflexive: x∈ X, x∼ x.

Symmetric: x, y∈ X x∼ y, y∼ x.

Transitive: x, y, z∈ X x∼ y y∼ z, x∼ z.

equivalence relation ? reflexive

. symmetric transitive

, ; A, B

, x A B . A a x a∼ x B

b x b∼ x. symmetric transitive a∼ b.

A B . A B

∼ X equivalence relation. x∈ X,

{y ∈ X : y ∼ x}, X x . , x

equivalence class, [x] . , [x] x

. reflexive , [x] , x∼ x,

x∈ [x]. y∼ x, [x] = [y]. , z∈ [x], z∼ x.

y∼ x, symmetric transitive z∼ y, z∈ [y]. [x]⊆ [y]. [y]⊆ [x],

[y] = [x]. y̸∼ x ( (y, x)̸∈ S), 前 /0

, [y]∩ [x] = /0. 言 , X equivalence relation∼, X

equivalence classes . X /∼ X

equivalence relation equivalence classes. X /∼ ∼

X . X subsets ,

X partition, .

Definition 4.2.2. X set, I index set. i∈ I, Ci X nonempty subset X =^∪

i∈I

C_i C_i∩Cj= /0, for i̸= j, {Ci: i∈ I} X partition.

X partition, X . 前 ,

X equivalence relation, equivalence relation equivalence classes (

) X partition. , X

partition X =^∪

i∈I

C_i, x, y∈ X, x∼ y x, y∈ Ci, for some i∈ I (

), , ∼ equivalence relation.

X =^∪

i∈I

Ci, x∈ X, i∈ I, x∈ Ci, x∼ x (

reflexive). x∼ y, x, y∈ Ci, for some i∈ I, y, x∈ Ci, y∼ x ( symmetric). x∼ y, y ∼ z, i, j∈ I x, y∈ Ci, y, z∈ Cj. y∈ Ci∩Cj,

i̸= j Ci∩Cj= /0 i = j. x, z∈ Ci, x∼ z ( transitive).

.

Theorem 4.2.3. X set.

(1) ∼ X equivalence relation, {[x] : [x] ∈ X/ ∼} X partition.

(2) I index set {Ci: i∈ I} X partition, x, y∈ X, x∼ y x, y∈ Ci, for some i∈ I, ∼ X equivalence relation.

Example 4.2.4. 數 Z 2 數 C1={2n : n ∈ Z}, 3 數

C2={3n : n ∈ Z} 5 數 C3={5n : n ∈ Z}, {C1,C2,C3}

Z partition. 7 2, 3 5 數 ( 7̸∈ C1∪C2∪C3),

Z ̸= C1∪C2∪C3. C1∩C2̸= /0, 6∈ C1∩C2. C1∩C3̸= /0, C2∩C3̸= /0.

Z 3 subset C1={n : n = 3m,m ∈ Z},C2={n : n = 3m + 1,m ∈ Z}

C3 ={n : n = 3m + 2,m ∈ Z}, {C1,C2,C3} Z partition. ,

C₁,C₂,C₃ 3 數 0, 1 2 .

Z = C1∪C2∪C3, C1∩C2= C1∩C3= C2∩C3= /0. partition, Z equivalence relation x∼ y x, y∈ Ci, for some i∈ {1,2,3}. x, y∈ C1

x = 3m, y = 3m^′ for some m, m^′∈ Z x− y = 3(m − m^′), 3| x − y ( 3 x− y). x, y∈ C2 x, y∈ C3, 3| x − y. equivalence relation

x∼ y 3| x − y. ∼ equivalence relation. x∈ Z,

3| x−x, x∼ x. x∼ y, 3| x−y, 3| −(x−y). 3| y−x, y∼ x. x∼ y y∼ x, 3| x −y 3| y−z. 3| (x −y)+(y −z), 3| x −z.

x∼ z. C1= [0] = [4], C2= [1] = [−3] C3= [2] = [11]... . Z/ ∼= {[0],[1],[2]}.

Question 4.4. 數 m, I ={0,1,...,m−1} index set. Z partition, Ci={mk + i : k ∈ Z}, i ∈ I. partition equivalence relation ?

equivalence relation partition ,

學 數學 論 . 前 .

數. .

Proposition 4.2.5. X finite set, equivalence relation equivalence classes C₁, . . . ,C_n. #(X ) #(C_i) 數,

#(X ) =

∑

#(C_i).

Proof. 前 i̸= j , C_i∩Cj= /0. C_i .

X Ci , X 數 C1, . . . ,Cn

數 . 

Example 4.2.6. A ={1,2,3} X =P(A). X relation,

B,C∈ X, B ∼ C #(B) = #(C). ∼ X equivalence relation.

, B∈ X, #(B) = #(B), B∼ B. B∼ C, #(B) = #(C),

4.3. Order Relation 61

#(C) = #(B), C∼ B. B∼ C C∼ D, #(B) = #(C) #(C) = #(D)

#(B) = #(D). B∼ D.

equivalence relation equivalence classes X =P(A) parti-

tion. partition:

: {/0}

: {{1},{2},{3}}.

: {{1,2},{1,3},{2,3}}.

: {{1,2,3}}.

equivalence classes 數 (₃

0

), (₃

1

), (₃

2

), (₃

3

). Proposition

4.2.5 (

3 0 )

+ (3

1 )

+ (3

2 )

+ (3

3 )

= #(X ) = #(P(A)) = 2³= 8.

Question 4.5. n 數, A ={1,2,...,n} X =P(A). X relation, B,C∈ X, B ∼ C #(B) = #(C). m∈ N 0 < m < n, {1,2,...,m}

equivalence class 數 ?

(n 0 )

+ (n

1 )

+··· + ( n

n− 1 )

+ (n

n )

= 2ⁿ.

4.3. Order Relation

數學 relation order relation, .

relation, relation ,

order relation.

relation , “ ”

. , ∼ ,

“ ”, “≼” .

Definition 4.3.1. X nonempty set ≼ X relation. ≼

, ≼ X partial order.

(1) x∈ X, x≼ x.

(2) x, y∈ X x≼ y y≼ x, x = y.

(3) x, y, z∈ X x≼ y y≼ z, x≼ z.

Definition 4.3.1 (1) reflexive , (3) transitive

. (2) symmetric . x̸= y,

x≼ y y≼ x. S⊆ X ×X relation, x̸= y ,

(x, y) (y, x) S , anti-symmetric. ≼ X

partial order, (X ,≼) poset.

Example 4.3.2. A nonempty set, X =P(A). X relation⊆, (X ,⊆) poset.

Question 4.6. A nonempty set, X =P(A). X relation ⊇,

(X ,⊇) poset?

Question 4.7. 數 R ≤, (R,≤) poset? (R,≥)

poset?

poset (X,≼) , x, y∈ X x≼ y y≼ x, x, y comparable

( ). Definition 4.3.1 “partial” order,

X comparable. A ={1,2} , ⊆ P(A)

partial order. {1},{2} ∈ P(A) comparable, {1} ⊆ {2} {2} ⊆ {1}

. 數 (R,≤) poset comparable .

order relation.

Definition 4.3.3. X nonempty set ≼ X relation. ≼

, ≼ X total order.

(1) x, y∈ X x≼ y y≼ x, x = y.

(2) x, y, z∈ X x≼ y y≼ z, x≼ z.

(3) x, y∈ X, x≼ y y≼ x.

Definition 4.3.3 (3) comparable, total

. (3) , reflexive, x, y ,

, x≼ x. total order partial order (

). ≼ X total order, (X ,≼) total ordered set.

total order linear order simple order.

Question 4.8. 數 R <, (R,<) total ordered set?

前 order≼ “ ”, x≼ x x = x.

數 ≤ < order ? , (X ,≼)

total ordered set, x≺ y x≼ y x̸= y. , ≺

X strict total order. .

Definition 4.3.4. X nonempty set ≺ X relation. ≺

, ≺ X strict total order.

(1) x, y, z∈ X x≺ y y≺ z, x≺ z.

(2) x, y∈ X, x = y, x≺ y y≺ x , .

Definition 4.3.4 , (2) trichotomy ( ).

4.3. Order Relation 63

Example 4.3.5. 數 C strict order.

a + bi, c + di∈ C, a, b, c, d ∈ R i² =−1. (a + bi)≺ (c + di) (1) a < c (2) a = c b < d. (C,≺) strict total ordered set.

transitive . a + bi, c + di, e + f i∈ C a, b, c, d, e, f ∈ R (a + bi)≺ (c + di)

(c + di)≺ (e + fi). ≺ , a≤ c c≤ e, a≤ e.

論: ( ) a < e, ≺ (a + bi)≺ (e + fi); ( ) a = e, a = c = e. (a + bi)≺ (c + di) b < d, (c + di)≺ (e + fi) d < f .

b < f , ≺ (a + bi)≺ (e + fi). ≺ transitive . ,

a + bi̸= c + di, 數 a̸= c b̸= c. a̸= c, 數 a < c

c < a, (a + bi)≺ (c + di) (c + di)≺ (a + bi). a = c, b̸= d,

數 (a + bi)≺ (c + di) (c + di)≺ (a + bi). 數 ≺

comparable, ≺ trichotomy .

(C,≺) strict total ordered set 數 < order.

數 ? . , 數 數

sets, . order

. 數 :

A: a < b, c a + c < b + c.

M: a < b, 0 < c ac < bc.

數 ≺ A. M.

0≺ i, M , 0× i ≺ i × i, 0≺ −1. ≺ , ≺

M.

, C strict total order ≺

數 A M. (C,≺) , ,

0≺ i i≺ 0 . 0≺ i, M 0≺ −1, 數

. i≺ 0, A i + (−i) ≺ 0 + (−i), 0≺ −i. M,

0× (−i) ≺ (−i) × (−i), 0≺ −1, 數 . C

strict total order , 數 .

Question 4.9. (C,≺) strict total ordered set Example 4.3.5 A

M, C z, 0≺ z².

strict total order total order. 前 , total ordered set (X ,≼), strict total order.

Proposition 4.3.6. (X ,≼) total ordered set. x≺ y x≼ y

x̸= y, , ≺ X strict total order.

Proof. transitive , x, y, z∈ X x≺ y y≺ z, x≺ z.

x≺ y x≼ y x̸= y, y≺ z y≼ z y̸= z. ≼ total order transitive

, x≼ z. x̸= z. , x = z, x≼ y z≼ y.

y≼ z, ≼ total order anti-symmetric , y = z. y≺ z ( y ̸= z) , x̸= z. x≺ z.

trichotomy . ≼ total order total , x, y∈ X,

x≼ y y≼ x. x = y, x, y x = y. x̸= y, x≼ y y≼ x,

x≺ y y≺ x. x, y x = y, x≺ y y≺ x. x, y x = y, x≺ y

y≺ x . x = y, ≺ x≺ y y≺ x . x̸= y

x≺ y, y ≺ x . x≺ y, y ≺ x , ≺

x≼ y y≼ x. anti-symmetric , x = y. x̸= y .

x≺ y, y ≺ x . 

, ≺ X strict total order, x, y∈ X, x≼ y

x = y x≺ y, ≼ X total order.

Question 4.10. X nonempty set ≺ X strict total order.

x, y∈ X, x≼ y x = y x≺ y, ≼ X total order.

, X total order strict total order,

. 論 total order , strict total order .

, ≼ total order, ≺ strict total order,

.

order , , . (X ,≼)

poset. X T , u∈ X T upper bound, T

t t≼ u. u∈ X T upper bound T upper bound u^′,

u≼ u^′, u T least upper bound. , l∈ X T lower

bound, T t l≼ t. l∈ X T lower bound

T lower bound l^′, l^′≼ l, l T greatest lower bound.

, poset (X,≼) nonempty subset upper bound lower

bound. upper bound lower bound, least upper bound greatest

lower bound . :

Example 4.3.7. (A) (R,≤) total ordered set. T ={x ∈ R : 0 < x < 1}.

1 數 T upper bound, 1 T least upper bound.

0 數 T lower bound, 0 T greatest lower bound. {x ∈ R : x ≥ 0}, upper bound. {x ∈ R : x < 1}, lower bound.

(B) (Q,≤) total ordered set. T ={x ∈ Q :√

2 < x <√

3}. √

3

數 T upper bound, √

2 數 T lower bound. T

least upper bound. u∈ Q T least upper bound, √

3 < u, √ 3

u 數 ( ), u^′∈ Q √

3 < u^′< u.

4.3. Order Relation 65

u^′ T upper bound u, u T least upper bound ,

T least upper bound. T greatest lower bound.

(C) nonempty set A, (P(A),⊆) poset. P(A) nonempty

subset T , A T upper bound, B∈ T, B⊆ A. /0 T

lower bound. T least upper bound , U = ^∪

B∈T

B T least

upper bound. B∈ T, B⊆ U, U T upper bound.

U^′∈ P(A) T upper bound, B∈ T, B⊆ U^′, Corollary 3.3.4, U = ^∪

B∈T

B⊆ U^′. U = ^∪

B∈T

B T least upper bound. A ={1,2,3,4} ,

T ={{1,2},{1,3}}. {1,2} ∪ {1,3} = {1,2,3} T least upper bound.

Question 4.11. nonempty set A, (P(A),⊆) poset. P(A) nonempty subset T , T greatest lower bound .

, poset (X,≼) nonempty subset T , least upper bound

, . u, u^′∈ X T least upper

bound, u least upper bound u^′ upper bound, u≼ u^′. u^′≼ u, partial order anti-symmetric u = u^′. T greatest lower bound

, . .

Proposition 4.3.8. (X ,≼) total ordered set T X nonempty subset. T

least upper bound , . T greatest lower bound , .

(X ,≼) total ordered set , least upper bound greatest lower bound

. u∈ X T least upper bound, x≺ u,

x T upper bound. x T upper bound u≼ x

( ≺ strict total order, , x≺ u u≼ x ).

論.

Proposition 4.3.9. (X ,≼) total ordered set, T X nonempty subset u∈ X T least upper bound. x∈ X x≺ u, t∈ T x≺ t.

Proof. , t∈ T x≺ t , t∈ T x≺ t.

≺ X strict total order, , x≺ t, t ≺ x x = t .

t∈ T x≺ t t∈ T t≼ x. x T upper bound.

u T least upper bound, u≼ x. x≺ u 前 .

t∈ T x≺ t. 

Question 4.12. (X ,≼) total ordered set, T X nonempty subset l∈ X

T greatest lower bound. x∈ X l≺ x, t∈ T t≺ x.

poset (X,≼) nonempty subset T . ,

poset . poset comparable,

T , maximal element of T , T

. µ ∈ T t∈ T µ ≺ t, µ T maximal element.

, greatest element of T ( maximum element), T

. g∈ T t∈ T t≼ g, g T greatest

element. . m∈ T t∈ T t≺ m, m T

minimal element. l∈ T t∈ T l≼ t, l T least element

( minimum element). , T upper bound lower bound

T , T maximal element, greatest element minimal element, least element

T . upper bound lower bound, ,

. .

Example 4.3.10. (A) (R,≤) total ordered set. T ={x ∈ R : 0 < x < 1}.

T maximal element. µ ∈ T 0 <µ < 1,

t = (µ + 1)/2, 0 < t < 1, t∈ T µ < t. T maximal element. T

minimal element. T^′={x ∈ R : 0 ≤ x ≤ 1}. 1 T^′

maximal element greatest element, 0 T^′ minimal element least element.

(B) A ={1,2,3} (P(A),⊆) poset. T ={{1},{1,2},{2,3},{1,2,3}}.

{1,2,3} T maximal element greatest element. {1} T minimal element

B∈ T B⊂ {1}. {1} T least element, {2,3} ∈ T

{2,3} {2,3} ⊆ {1}. {2,3} T minimal element, B∈ T

B⊂ {2,3}. T^′={{1},{1,2},{2,3}}. T^′ greatest element,

{1,2} {2,3} T^′ maximal element. {2,3} T^′

maximal element minimal element.

Example 4.3.10 maximal element minimal element .

greatest element least element , . .

Proposition 4.3.11. (X ,≼) poset, T nonempty subset T greatest

element . T greatest element . T maximal element

, T maximal element T greatest element, T least upper bound.

Proof. , greatest element . g, g^′∈ T T greatest

element g̸= g^′. g^′∈ T g T greatest element, g^′≼ g.

g≼ g^′. ≼ partial order anti-symmetric , g^′≼ g g≼ g^′ g = g^′

. .

g∈ T T greatest element. t∈ T g≼ t, reflexive g = t.

言 , t∈ T g≺ t. g T maximal element. T maximal

element . µ ∈ T T maximal element. µ ∈ T, µ ≼ g.

maximal element g∈ T µ ≺ g , µ = g.

4.3. Order Relation 67

T maximal element T greatest element g, T maximal

element .

greatest element , t∈ T t≼ g, g T upper bound.

T upper bound u, g∈ T, upper bound , g≼ u, g

T least upper bound. 

Question 4.13. (X ,≼) poset, T nonempty subset T least element

. T least element , T minimal element .

T minimal element T least element, T greatest lower bound.

Question 4.14. (X ,≼) poset T nonempty subset. T least upper

bound u u∈ T T greatest element . , T greatest

lower bound l l∈ T T least element .

(X ,≼) poset, T nonempty subset, Proposition 4.3.11 Question 4.13 T greatest element T maximal element greatest element,

T least element T minimal element least element. (X ,≼)

partial ordered set total ordered set , comparable

maximal element greatest element minimal element least element.

(X ,≼) total ordered set maximal element greatest element , minimal element least element .

Proposition 4.3.12. (X ,≼) total ordered set, T nonempty subset. T maximal element , T maximal element T greatest element.

Proof. µ ∈ T T maximal element. t∈ T, µ ≺ t ,

t≼µ. µ T greatest element. 

Question 4.15. (X ,≼) total ordered set, T nonempty subset. T minimal element , T minimal element T least element.

, well order, .

Definition 4.3.13. (X ,≼) total ordered set. X nonempty subset T least element , (X ,≼) well-ordered set.

數 , well-ordered set.

數 , well-ordered set. 數

least element.

數 學 , Well-ordering

Theorem . nonempty set X, total order

≼ (X ,≼) well-ordered set.

Example 4.3.14. 數 Z, well-

ordered set. total order ≼ (Z,≼) well-ordered set.

relation: a, b∈ Z, a≼ b (1) |a| < |b| (2) |a| = |b| a≤ b.

(Z,≼) total ordered set. a≼ b b≼ a, |a| = |b| ( |a| < |b|

|b| < |a| ) a≤ b b≤ a, a = b, ≼ anti-symmetric .

a≼ b b≼ c, |a| ≤ |b| |b| ≤ |c|, |a| ≤ |c|. |a| < |c| a≼ c.

|a| = |c|, |a| = |b|, a≼ b a≤ b. |b| = |c|, b≼ c

b≤ c. |a| = |c| a≤ c, a≼ c. ≼ transitive

. total , a, b∈ Z |a|,|b| , |a| = |b|,

a, b , a, b∈ Z comparable, ≼ total . ,

數

0≺ −1 ≺ 1 ≺ −2 ≺ 2··· .

(Z,≼) total ordered set well-ordered set. Z nonempty

subset T , T . , ≼

T least element. , T

least element. ≼ order , T least element. (Z,≼) well-ordered set.

Question 4.16. Z relation: a, b∈ Z a≼ b (1)

ab≥ 0 |a| ≤ |b| (2) ab < 0 a≤ b. 數

0≺ −1 ≺ −2 ≺ −3··· ≺ 1 ≺ 2 ≺ 3··· .

(Z,≼) total ordered set. (Z,≼) well-ordered set?

Well-ordering Theorem Zorn’s Lemma Axiom of Choice ,

function 論.