數學導論
學數學
前言
學
學
數學
.
論
(Set)
.
數學
數 (Function)
論
學數學
,
學
(Logic),
論.
學
學
.
.
,
學
.
數學
,
,
.
,
,
,
.
.
.
,
,
論
.
v
Chapter 4
Relation and Order
relation. Relation
relation
relation.
set
relation,
equivalence relation. Equivalence relation
,
set
.
relation,
relation.
equivalence relation
relation
學
.
order. Order
set
(
),
.
4.1. Relation
sets X,Y .
from X to Y .
X ×Y
S
X ×X
,
relation S
nonempty subset,
nonempty subset S
x∼y
,
x∼y
x
S
X
relation.
Question 4.1.
x, y ∈ X
X
1 ∼ 2, 0 ∼ 1
relation
1 ∼ 0, 1 ∼ −1
relation
,
S
−1 ∼ 2.
relation.
X
0 ∼ −1.
nonempty set.
X
relation S.
S = X ×X
x ∼ y.
relation
Example 4.1.1(1)
X,Y
,
數
f (x) = x2 + 1
relation,
X
.
S = {(x, y) ∈ X ×Y : y = x2 + 1}.
S = {(x, x′ ) ∈ X × X : x > x′ }.
X = {1, 0, −1}.
(2)
.
(x, y) ∈ S
,
X = {1, 0, −1}, Y = {0, 1, 2}.
Y
.
S
relation
relation
Example 4.1.1. (1)
relation on X.
(x, y)
,
y
relation
S
,
relation.
Example 4.1.1(2)
relation.
55
56
4. Relation and Order
X ×Y
relation
,
.
X,Y
X ×Y
S
S
,
X,Y
.
relation
論
Example 4.1.1
(
數,
)
relation
X
.
.
S
.
Example 4.1.2.
X
“
”
?
relation
set P(X)
,
X
X
S ⊆ P(X) × P(X)
relation.
(A, B) ∈ S.
A⊆B
S = {(A, B) ∈ P(X) × P(X) : A ⊆ B}.
A∼B
A⊆B
relation.
論
,
Reflexive:
relation
.
,
on X,
power
relation
S
relation.
x∈X
S
x ∼ x,
(x, x) ∈ S, ∀ x ∈ X,
relation
Symmetric:
reflexive.
x, y ∈ X
S
x ∼ y,
y ∼ x,
(x, y) ∈ S ⇒ (y, x) ∈ S,
relation
Transitive:
symmetric.
x, y, z ∈ X
S
x∼y
y ∼ z,
x ∼ z,
((x, y) ∈ S) ∧ ((y, z) ∈ S) ⇒ (x, z) ∈ S,
relation
transitive.
,
(1) S
.
x∈X
reflexive
(x, x) ∈ S
(2) S
言
x = y.
x, (x, x)
X
x ̸= y
x∈X
S.
(x, y) ∈ S
reflexive.
reflexive,
(x, y)
(x, x)
,
S
,
S
.
(x, y) ∈ S
symmetric
(x, y)
(y, x)
symmetric.
S
(x, y) ∈ S
(y, x)
言
(y, x) ̸∈ S
S.
,
symmetric,
S
.
symmetric,
symmetric.
(3) S
transitive
(x, y), (y, z)
z∈X
(x, y)
(x, z)
(y, z) ∈ S.
(y, z)
S
,
(x, z)
S.
(x, y) ∈ S
transitive.
S
S
(x, y)
S
transitive.
言
4.1. Relation
,
57
(x, y), (y, z) ∈ S
transitive,
S
.
transitive,
transitive.
X = {1, 2, 3},
Example 4.1.3.
(x, z) ̸∈ S
relation S ⊆ X × X
reflexive, symmetric
transitive.
S = {(1, 1), (2, 2)},
(1)
S = {(1, 1), (2, 2), (3, 3), (1, 2)},
x ̸= y
S = {(1, 1), (2, 2), (1, 2)},
(1, 2) ∈ S
S = {(1, 1), (2, 2), (1, 2)},
sitive.
(2, 1) ̸∈ S,
transitive.
S
symmetric.
reflexive,
S
symmetric,
tran-
(1, 2), (2, 3) ∈ S
(1, 3) ̸∈ S,
S = {(1, 1), (2, 2), (1, 2), (2, 3), (1, 3)})
(1, 3) (
S
(1, 2),
symmetric.
S
S = {(1, 1), (2, 2), (1, 2), (2, 3)},
(2, 3),
S
symmetric.
S
S = {(1, 1), (2, 2), (1, 2), (2, 1)})
(2, 1) (
(3)
.
reflexive,
S
(x, y)
S
reflexive
S = {(1, 1), (2, 2)},
(2)
(3, 3) ̸∈ S.
reflexive.
S
(1, 2) ∈ S)
(
3∈X
reflexive,
S
transitive.
Example
reflexive, symmetric
.
.
reflexive
symmetric
論
Example 4.1.4.
symmetric
x ∼ y,
transitive
x ∼ x.
reflexive
S
(3, 3) ̸∈ S,
,
,
transitive
,
transitive,
transitive
,
symmetric
transitive
(3, 3)
S
(1, 1),
.
reflexive.
S
reflexive
relation on X
.
x ∼ y,
(
relation on X
.
x ∼ x.
S = {(1, 1), (2, 2), (1, 2), (2, 1)},
symmetric
S
3
X = {1, 2, 3}
x ∈ X,
1
2 ∼ 1),
symmetric
Question 4.2.
y ∼ x,
y∈X
,
transitive.
(
1
S
x∼y
x ∈ X,
,
X = {1, 2, 3}
.
symmetric
.
symmetric
,
,
x∼x
2
,
reflexive.
x
S
S
?
論
S
X
(2, 2)
relation.
X
y ∼ x.
,
x ∼ y,
.
x ∼ x.
,
論
1 ∼ 2),
S
symmetric,
S
transitive.
transitive.
reflexive?
S
symmetric
reflexive.
set,
X
,
,
transitive
transitive,
y∈X
reflexive
relation
relation
.
relation
relation
,
transitive
symmetric
reflexive
58
4. Relation and Order
Example 4.1.2
Example 4.1.5.
P(X)
relation
(A, A) ∈ S.
Proposition 3.1.4(1)),
(A, B) ∈ S
0/ ∈ P(X)
.
(X, 0)
/ ̸∈ S.
P(X)
symmetric.
.
A = 0/
(0,
/ X) ∈ S.
B=X
X ̸= 0,
/
X * 0,
/
symmetric.
S = {(A, B) ∈ P(X) × P(X) : A ∪ B = X}
nonempty set.
X
A⊆C (
S
(B, A) ̸∈ S
0/ ⊆ X,
S
B ⊆ C,
transitive.
(A, B) ∈ S
X ∈ P(X)
S
Question 4.3.
A⊆B
S
A, B ∈ P(X)
,
reflexive.
S
(B,C) ∈ S,
(A,C) ∈ S,
Proposition 3.1.4(2)).
A ∈ P(X),
reflexive.
S
transitive.
S = {(A, B) ∈ P(X) × P(X) : A ⊆ B}
nonempty set.
X
relation.
A⊆A (
.
relation.
(1)
S
reflexive
(2)
S
symmetric
(3)
S
transitive
?
,
?
,
?
reflexive
S
,
?
symmetric
S
transitive
S
?
?
4.2. Equivalence Relation
relation
,
equivalence relation.
,
equivalence classes,
,
數學
.
cd,
.
,
.
cd
?
,
數學,
.
.
,
,
;
;
,
,
X
x∼y
.
.
,
,前
x∈X
:
x ∼ x.
,
,
symmetric.
reflexive.
x∼y
y∼z
X
relation S
,
x ∼ z.
relation
reflexive, symmetric
relation
x ∼ y,
(
.
transitive
relation
relation
relation
X ×X
.
subset
:
y ∼ x.
:
x∼y
,
,
relation
,
relation
,
x, y
transitive.
transitive
),
x, y
reflexive, symmetric
.
,
X
,
∼
X
4.2. Equivalence Relation
59
∼
Definition 4.2.1.
relation,
X
,
relation
equivalence relation
x ∈ X,
Reflexive:
x ∼ x.
Symmetric:
x, y ∈ X
x ∼ y,
y ∼ x.
Transitive:
x, y, z ∈ X
x∼y
y ∼ z,
equivalence relation
x ∼ z.
?
.
symmetric
reflexive
transitive
,
,
x
;
A
B
b
x
A
A
a
b ∼ x.
symmetric
.
B
∼
X
equivalence class,
reflexive
symmetric
z ∼ y,
y ̸∼ x (
[y] ∩ [x] = 0.
/
,
言
0/
X
X
X/ ∼
∼
subsets
,
.
X
subset
Ci ∩C j = 0,
/ for i ̸= j,
Ci
前
X
Definition 4.2.2.
X=
[y] ⊆ [x],
X/ ∼
partition,
∪
[x] ⊆ [y].
.
.
X
z ∼ x.
equivalence relation ∼,
equivalence classes.
X
z ∈ [x],
X
equivalence classes
equivalence relation
x ∼ x,
z ∈ [y].
(y, x) ̸∈ S),
,
x
x
,
[x] = [y].
transitive
[y] = [x].
,
,
[x]
y ∼ x,
y ∼ x,
x ∈ X,
, [x]
,
x ∈ [x].
B
.
x
B
a ∼ b.
transitive
A
.
[x]
a∼x
x
equivalence relation.
X
{y ∈ X : y ∼ x},
.
.
A, B
set, I
i ∈ I, Ci
index set.
{Ci : i ∈ I}
nonempty
X
partition.
X
i∈I
X
X
partition,
equivalence relation,
partition X =
∪
equivalence relation
)
X=
∪
),
x ∼ y,
reflexive).
symmetric).
i ̸= j
x ∈ X,
i = j.
i, j ∈ I
x, z ∈ Ci ,
X
x, y ∈ Ci , for some i ∈ I (
∼
equivalence relation.
i ∈ I,
x ∈ Ci ,
x, y ∈ Ci , for some i ∈ I,
x ∼ y, y ∼ z,
Ci ∩C j = 0/
.
,
Ci ,
i∈I
,
x∼y
i∈I
,
equivalence classes (
partition.
X
x, y ∈ X,
Ci ,
. 前
X
y, x ∈ Ci ,
x, y ∈ Ci , y, z ∈ C j .
x∼z (
x∼x (
y∼x (
y ∈ Ci ∩C j ,
transitive).
60
4. Relation and Order
Theorem 4.2.3.
set.
X
(1)
∼
X
equivalence relation,
{[x] : [x] ∈ X/ ∼}
X
(2)
I
index set
{Ci : i ∈ I}
partition,
x, y ∈ X,
X
x, y ∈ Ci , for some i ∈ I,
數Z
Example 4.2.4.
partition.
Z ̸= C1 ∪C2 ∪C3 .
7
C1 ,C2 ,C3
3
Z = C1 ∪C2 ∪C3 ,
6 ∈ C1 ∩C2 .
C1 ∩C3 ̸= 0,
/ C2 ∩C3 ̸= 0.
/
Z
數
2
0, 1
partition.
x∼y
x − y).
for some
x, y ∈ C2
x∼y
m, m′
x, y ∈ C3 ,
x∼y
3 | x−y (
3 | x − y.
equivalence relation
∼
x ∼ y,
y ∼ x,
x ∼ z.
x, y ∈ C1
x − y = 3(m − m′ ),
∈Z
x ∼ x.
Z
partition,
x, y ∈ Ci , for some i ∈ {1, 2, 3}.
3 | x − y.
3 | x − x,
,
.
C1 ∩C2 = C1 ∩C3 = C2 ∩C3 = 0.
/
= 3m, y = 3m′
{C1 ,C2 ,C3 }
7 ̸∈ C1 ∪ C2 ∪ C3 ),
數(
5
{C1 ,C2 ,C3 }
equivalence relation
y ∼ x.
2, 3
數
subset C1 = {n : n = 3m, m ∈ Z},C2 = {n : n = 3m + 1, m ∈ Z}
3
C3 = {n : n = 3m + 2, m ∈ Z},
x
C3 = {5n : n ∈ Z},
C1 ∩C2 ̸= 0,
/
Z
C1 = {2n : n ∈ Z}, 3
數
5
x∼y
equivalence relation.
X
數
2
C2 = {3n : n ∈ Z}
Z
∼
partition.
x ∈ Z,
equivalence relation.
3 | x − y,
3 | x−y
3
3 | −(x − y).
3 | y − z.
3 | y − x,
3 | (x − y) + (y − z),
C1 = [0] = [4], C2 = [1] = [−3]
3 | x − z.
.
C3 = [2] = [11]...
Z/ ∼= {[0], [1], [2]}.
Question 4.4.
I = {0, 1, . . . , m − 1}
數 m,
Ci = {mk + i : k ∈ Z}, i ∈ I.
partition
?
partition
數學
.
論
,
.
前
.
數.
Proposition 4.2.5.
partition,
Z
equivalence relation
equivalence relation
學
index set.
finite set,
X
equivalence classes C1 , . . . ,Cn .
#(X)
equivalence relation
數,
#(Ci )
n
#(X) = ∑ #(Ci ).
i=1
Proof.
i ̸= j
前
X
數
Ci
, Ci ∩C j = 0.
/
,
.
Ci
數
X
C1 , . . . ,Cn
.
Example 4.2.6.
A = {1, 2, 3}
B,C ∈ X, B ∼ C
,
X = P(A).
#(B) = #(C).
B ∈ X,
#(B) = #(B),
relation,
X
∼
B ∼ B.
X
equivalence relation.
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
tion.
X = P(A)
equivalence classes
parti-
partition:
: {0}
/
: {{1}, {2}, {3}}.
: {{1, 2}, {1, 3}, {2, 3}}.
: {{1, 2, 3}}.
equivalence classes
(3) (3) (3) (3)
0 , 1 , 2 , 3 .
數
4.2.5
( ) ( ) ( ) ( )
3
3
3
3
+
+
+
= #(X) = #(P(A)) = 23 = 8.
0
1
2
3
Question 4.5.
n
數, A = {1, 2, . . . , n}
B,C ∈ X, B ∼ C
equivalence class
X = P(A).
m∈N
#(B) = #(C).
Proposition
relation,
X
{1, 2, . . . , m}
0 < m < n,
數
?
( ) ( )
(
) ( )
n
n
n
n
+
+···+
+
= 2n .
0
1
n−1
n
4.3. Order Relation
relation
數學
relation,
order relation,
.
relation
,
order relation.
relation
,
.
∼
,
“
”,
“≼”
Definition 4.3.1.
,
≼
(1)
x ∈ X,
(2)
x, y ∈ X
(3)
x, y, z ∈ X
X
(2)
x≼y
y ≼ x.
(x, y)
(y, x)
partial order,
,
≼
relation.
X
≼
partial order.
x ≼ x.
x≼y
y ≼ x,
x≼y
Definition 4.3.1
.
”
.
nonempty set
X
“
y ≼ z,
(1)
x ≼ z.
reflexive
symmetric
S ⊆ X ×X
S
x = y.
,
(3)
x ̸= y,
.
x ̸= y
relation,
,
anti-symmetric.
(X, ≼)
transitive
poset.
≼
X
,
62
4. Relation and Order
Example 4.3.2.
relation ⊆,
X = P(A).
nonempty set,
A
(X, ⊆)
X
poset.
Question 4.6.
(X, ⊇)
X = P(A).
nonempty set,
A
relation ⊇,
X
poset?
數R
Question 4.7.
≤,
(R, ≤)
(R, ≥)
poset?
poset?
poset (X, ≼)
(
x, y ∈ X
,
x≼y
y ≼ x,
). Definition 4.3.1
partial order.
A = {1, 2}
{1}, {2} ∈ P(A)
comparable,
數 (R, ≤)
.
“partial” order,
comparable.
X
comparable
x, y
poset
⊆
,
{1} ⊆ {2}
P(A)
{2} ⊆ {1}
comparable
.
order relation.
Definition 4.3.3.
≼
,
x, y ∈ X
(2)
x, y, z ∈ X
x≼y
y ≼ x.
comparable,
,
x ≼ x.
≼
).
x ≼ z.
(3)
(3)
,
total order
x, y
total order
partial order (
(X, ≼)
linear order
<,
order ≼
前
≤
數
“
x≺y
Definition 4.3.4.
≺
,
(1)
(2)
X
x≼x
x≼y
?
x = x.
(X, ≼)
,
x ̸= y.
,
≺
.
≺
nonempty set
X
relation.
≺
strict total order.
X
x, y, z ∈ X
x≺y
y ≺ z,
x, y ∈ X,
Definition 4.3.4
total ordered set?
”,
order
strict total order.
X
total ordered set.
(R, <)
<
total ordered set,
,
simple order.
數R
Question 4.8.
total
reflexive,
total order,
X
≼
x = y.
y ≼ z,
x≼y
Definition 4.3.3
.
y ≼ x,
x≼y
x, y ∈ X,
(3)
relation.
X
total order.
X
(1)
≼
nonempty set
X
x ≺ z.
x = y, x ≺ y
,
(2)
y≺x
trichotomy (
,
).
.
4.3. Order Relation
63
Example 4.3.5.
a + bi, c + di ∈ C,
(1) a < c
a, b, c, d ∈ R
(2) a = c
transitive
i2
(a + bi) ≺ (c + di)
(C, ≺)
b < d.
(c + di) ≺ (e + f i).
≺
論: (
strict total ordered set.
a≤c
≺
a < e,
≺
(a + bi) ≺ (c + di)
c < a,
≺
comparable,
(C, ≺)
transitive
b ̸= c.
a ̸= c,
(c + di) ≺ (a + bi).
(a + bi) ≺ (c + di)
數
a = e,
d < f.
.
,
數
a<c
a = c,
b ̸= d,
≺
數
.
strict total ordered set
數
?
.
sets,
,
.
order.
<
數
數
order
:
數
A:
a < b,
c
M:
a < b,
0<c
ac < bc.
≺
A.
數
0 ≺ i,
)
(c + di) ≺ (a + bi).
trichotomy
數
.
a ≤ e.
(c + di) ≺ (e + f i)
≺
a ̸= c
數
c ≤ e,
b < d,
(a + bi) ≺ (e + f i).
a + bi ̸= c + di,
(a + bi) ≺ (c + di)
(a + bi) ≺ (e + f i); (
(a + bi) ≺ (c + di)
a = c = e.
a, b, c, d, e, f ∈ R
,
)
strict order.
= −1.
a + bi, c + di, e + f i ∈ C
.
b < f,
C
數
M
a + c < b + c.
M.
0 × i ≺ i × i,
,
0 ≺ −1.
≺
≺
,
M.
C
,
strict total order ≺
A
數
0≺i
i≺0
0 ≺ i,
.
i ≺ 0,
.
(C, ≺)
M.
0 ≺ −1,
z,
strict total order
(X, ≼),
Example 4.3.5
A
z2 .
total order.
,
前
total ordered set
strict total order.
(X, ≼)
Proposition 4.3.6.
x ̸= y,
,≺
Proof.
transitive
x≺y
0≺
C
.
數
strict total ordered set
C
M,
M,
.
,
(C, ≺)
Question 4.9.
數
0 ≺ −i.
數
strict total order
,
0 ≺ −1,
M
i + (−i) ≺ 0 + (−i),
A
0 × (−i) ≺ (−i) × (−i),
,
x≼y
x ̸= y,
total ordered set.
x≺y
x≼y
y ≺ z,
x ≺ z.
strict total order.
X
y≺z
,
x, y, z ∈ X
y≼z
y ̸= z.
x≺y
≼
total order
transitive
64
4. Relation and Order
x ≼ z.
,
y ≼ z,
x ̸= z.
≼
total order
x ̸= z.
,
x≺y
,
≼
.
y ≺ x.
x ≺ y, y ≺ x
,
y ≼ x,
x = y, x ≺ y
y≺x
x ̸= y
.
≺
,
x ̸= y
x = y.
.
.
≺
,
x ≺ y,
≼
x≼y
≼
≼
total order.
X
total order
total order
,
strict total order.
X
x ≺ y,
x=y
X
論
≺
nonempty set
X
x≼y
total order.
X
,
x, y ∈ X,
strict total order,
X
Question 4.10.
.
x≼y
x ≺ y, y ≺ x
.
anti-symmetric
x, y ∈ X,
x, y
x≺y
x ≺ y, y ≺ x
x, y ∈ X,
y ̸= z)
,
x ̸= y,
≺
x = y,
y ≼ x.
total
x = y.
x, y
x = y, x ≺ y
x, y
.
x=y
y≺z (
y = z.
total order
x = y,
y ≺ x.
x≼y
z ≼ y.
x ≺ z.
y ≼ x.
y≺x
x≼y
x = z,
anti-symmetric
trichotomy
x≼y
,
strict total order,
,
strict total order
≺
total order,
.
strict total order,
.
order
poset.
,
t ≼ u.
u ≼ u′ ,
u∈X
u
T
lower bound
l′
upper bound
lower bound
數
upper bound.
0
T
{x ∈ R : x < 1},
total ordered set.
upper bound,
least upper bound.
u
upper bound
least upper bound
total ordered set.
upper bound,
T
(Q, ≤)
T
lower bound
greatest lower bound.
T
lower bound,
(R, ≤)
lower bound,
T
T
lower
T
lower
greatest
:
數
1
l∈X
nonempty subset
.
Example 4.3.7. (A)
數
l
poset (X, ≼)
bound.
(B)
≼ l,
l∈X
T
upper bound u′ ,
T
,
l ≼ t.
t
l′,
,
upper bound
T
(X, ≼)
.
upper bound,
T
least upper bound.
T
bound,
0
u∈X
T,
X
t
T
,
數(
u∈Q
1
T
T = {x ∈ R : 0 < x < 1}.
least upper bound.
greatest lower bound.
lower bound.
T = {x ∈ Q :
√
2
數
√
√
2 < x < 3}.
{x ∈ R : x ≥ 0},
√
3
lower bound.
T
√
√
T
least upper bound,
3 < u,
3
√
′
′
),
u ∈Q
3 < u < u.
T
4.3. Order Relation
u′
65
upper bound
T
u,
least upper bound.
T
(C)
(P(A), ⊆)
lower bound.
B ∈ T,
upper bound.
U′
∈ P(A)
U=
∪
B ∈ T,
least upper bound
T
B⊆U .
∪
U=
B∈T
least upper bound
∈X
u≼
least upper
T
u′ .
u′ ≼ u,
greatest lower bound
T
.
(X, ≼)
least upper bound
total ordered set
,
.
total ordered set
T
, least upper bound
upper bound.
T
, x≺u
T
,
.
greatest lower bound
x ≺ u,
least upper bound,
T
x
strict total order,
nonempty subset.
X
greatest lower bound
T
u∈X
.
≺
.
u = u′ .
anti-symmetric
Proposition 4.3.8.
T
,
P(A)
poset.
upper bound,
.
(X, ≼)
A = {1, 2, 3, 4}
u, u′
u′
least upper bound
,
least
least upper bound.
T
.
partial order
T
Corollary 3.3.4,
nonempty subset T ,
,
(
B
greatest lower bound
T
poset (X, ≼)
u
T
upper bound.
T
⊆ U ′,
(P(A), ⊆)
nonempty set A,
,
x
U
{1, 2} ∪ {1, 3} = {1, 2, 3}
nonempty subset T ,
bound,
B
0/
B∈T
least upper bound.
T
∪
B∈T
T = {{1, 2}, {1, 3}}.
Question 4.11.
U=
B ∈ T,
B
nonempty
B ⊆ A.
,
B ⊆ U,
upper bound,
T
′
,
P(A)
poset.
upper bound,
T
least upper bound
T
greatest lower bound.
T
nonempty set A,
subset T , A
u
u≼x
upper bound
u≼x
).
論.
(X, ≼)
Proposition 4.3.9.
least upper bound.
T
Proof.
,
≺
x≺t
T
x ≺ u,
t∈T
x≺t
Question 4.12.
t ∈T
u∈X
nonempty subset
t ∈T
x ≺ t.
t∈T
,
t ≼ x.
x
u ≼ x.
x ≺ t.
.
x=t
T
upper bound.
x≺u
.
前
x ≺ t.
(X, ≼)
total ordered set, T
greatest lower bound.
x∈X
poset (X, ≼)
poset
X
, x ≺ t, t ≺ x
least upper bound,
t ∈T
T
x∈X
strict total order,
X
t ∈T
u
total ordered set, T
X
l ≺ x,
nonempty subset T
.
poset
nonempty subset
t ∈T
l∈X
t ≺ x.
.
,
comparable,
66
4. Relation and Order
,
T
maximal element of T ,
µ ∈T
.
,
greatest element of T (
t∈T
l∈T
,
T
.
T
,T
upper bound
.
t = (µ + 1)/2,
(R, ≤)
least element
T
lower bound
,
{1, 2, 3}
µ < t.
greatest element,
B∈T
{2, 3}
{2, 3} ⊆ {1}.
T′
maximal element
minimal element
T
{2, 3} ∈ T
least element,
T
B∈T
minimal element,
T
T′
greatest element,
T′
{2, 3}
minimal element.
greatest element
maximal element
least element
.
maximal element
Proof.
,
g ̸= g′ .
g
greatest element,
partial order
least upper bound.
T
g, g′ ∈ T
.
anti-symmetric
g′ ≼ g
,
T
greatest
g′ ≼ g.
greatest element,
T
greatest
T
maximal element
T
greatest element
g′ ∈ T
.
nonempty subset
.
T
.
.
poset, T
greatest element
T
minimal element
,
(X, ≼)
Proposition 4.3.11.
.
{1}
maximal element.
Example 4.3.10
≼
least element.
T = {{1}, {1, 2}, {2, 3}, {1, 2, 3}}.
poset.
{1}
T
T′
1
minimal element
T ′ = {{1}, {1, 2}, {2, 3}}.
{2, 3}
T
T′
0
{2, 3}
B ⊂ {2, 3}.
element
maximal element.
T
greatest element.
B ⊂ {1}.
0 < µ < 1,
= {x ∈ R : 0 ≤ x ≤ 1}.
(P(A), ⊆)
maximal element
T
T = {x ∈ R : 0 < x < 1}.
µ ∈T
T′
A = {1, 2, 3}
(B)
g ≼ g′
g = g′
.
g∈T
言
T
minimal element, least element
total ordered set.
t∈T
0 < t < 1,
maximal element
g ≼ g′ .
l
m
lower bound,
minimal element.
element
greatest
T
t ≺ m,
upper bound
maximal element.
T
,
g
.
Example 4.3.10. (A)
{1, 2}
l ≼ t,
maximal element, greatest element
T
T
t ∈T
t∈T
minimum element).
maximal element.
T
t ≼ g,
m∈T
.
minimal element.
µ
maximum element),
g∈T
element.
(
µ ≺ t,
t∈T
.
T
T
t ∈T
,
element
maximal element
.
t ∈T
greatest element.
g ≺ t.
µ ∈T
g∈T
T
g
T
g ≼ t,
reflexive
maximal element.
maximal element.
µ ≺g
µ ∈ T,
,
T
µ ≼ g.
µ = g.
g = t.
maximal
4.3. Order Relation
67
maximal element
T
element
greatest element g,
T
maximal
T
.
greatest element
t∈T
,
g ∈ T,
upper bound u,
T
t ≼ g,
g
upper bound.
T
upper bound
g ≼ u,
,
g
least upper bound.
T
(X, ≼)
Question 4.13.
.
least element
T
poset
u∈T
bound u
l∈T
lower bound l
(X, ≼)
poset, T
4.13
greatest element
T
least element
(X, ≼)
total ordered set
minimal element
maximal element
Proof.
,
maximal element.
µ
T
greatest element.
(X, ≼)
minimal element
,
(X, ≼)
least element
數
,
,
數
nonempty subset.
T
T
least element.
total ordered set.
(X, ≼)
,
well order,
,
T
greatest element.
t ∈ T, µ ≺ t
minimal element
,
Definition 4.3.13.
,
nonempty subset.
T
total ordered set, T
T
least element.
greatest element
total ordered set, T
T
Question 4.15.
(X, ≼)
.
µ ∈T
t ≼ µ.
greatest element,
comparable
maximal element
T
Question
least element.
maximal element
(X, ≼)
Proposition 4.3.12.
Proposition 4.3.11
minimal element
least element
greatest
T
.
,
greatest element
total ordered set
,
maximal element
T
minimal element
T
partial ordered set
maximal element
nonempty subset,
least upper
T
.
least element
T
greatest lower bound.
T
greatest element
T
.
nonempty subset.
T
least element
T
minimal element
T
least element,
T
(X, ≼)
Question 4.14.
nonempty subset
,
minimal element
T
T
poset, T
.
X
nonempty subset T
well-ordered set.
well-ordered set.
well-ordered set.
數
least element.
,
數學
Theorem
≼
(X, ≼)
.
nonempty set X,
well-ordered set.
Well-ordering
total order
68
4. Relation and Order
Example 4.3.14.
total order ≼
ordered set.
a, b ∈ Z,
relation:
(Z, ≼)
a≼b
b ≼ c,
|a| ≤ |b|
|a| = |c|,
|a| = |b|,
b ≤ c.
|a| = |c|
.
total
b ≤ a,
|b| ≤ |c|,
well-ordered set.
(2) |a| = |b|
b ≼ a,
a ≼ c.
comparable,
≼
b≼c
≼
,
total
.
a ≼ c.
|b| = |c|,
|a|, |b|
a, b ∈ Z
|a| < |c|
a ≤ b.
a ≤ c,
|a| < |b|
anti-symmetric
|a| ≤ |c|.
a≼b
a ≤ b.
|a| = |b| (
≼
a = b,
a, b ∈ Z
,
,
a, b
(Z, ≼)
a≼b
a≤b
)
well-
(1) |a| < |b|
total ordered set.
|b| < |a|
a≼b
Z,
數
transitive
|a| = |b|,
.
,
數
0 ≺ −1 ≺ 1 ≺ −2 ≺ 2 · · · .
(Z, ≼)
subset T ,
T
total ordered set
.
T
≼
nonempty
≼
,
least element.
least element.
Z
well-ordered set.
,
order
,T
least element.
T
(Z, ≼)
well-ordered
set.
Question 4.16.
ab ≥ 0
|a| ≤ |b|
Z
relation:
(2) ab < 0
a, b ∈ Z
a ≤ b.
a≼b
(1)
數
0 ≺ −1 ≺ −2 ≺ −3 · · · ≺ 1 ≺ 2 ≺ 3 · · · .
(Z, ≼)
Well-ordering Theorem
function
total ordered set.
Zorn’s Lemma
論.
(Z, ≼)
well-ordered set?
Axiom of Choice
,