tài liệu đại số tuyến tính

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§ λ
λ
λ
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λ J − λI
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§ C
C
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“2 + 2 = 4”
“π ”
p q

p p
¬p p p “2 + 2 = 4”
¬(2 + 2 = 4) “2 + 2 = 4” p

“π ” p ¬p
“π ”
p q p q p ∧ q
p q
(2 + 2 = 4) ∧ (π ) (2 + 2 =
4) ∧ (π )
p q p q p ∨ q
p “ ” q p q
“ ” “p q”
p q
p q (2 + 2 = 4) ∨(π )
p q p q
p → q p q
(2 + 2 = 4) → (π )
(π ) → (2 + 2 = 4)
p q q
p
⊲ p q
⊲ p q
⊲ p q
p q p
q
⊲ q q
⊲ q p
⊲ q p

§
p → q p
q
p q

p q p q
p ↔ q p q
(2 + 2 = 4) ↔ (π )
p → q p ↔ q
p q
⊲ p q
⊲ p q
P Q
P ↔ Q
P ⇔ Q
p q r
¬(¬p) ⇐⇒ p
p ∧ q ⇐⇒ q ∧ p, p ∨q ⇐⇒ q ∨ p.
(p ∧ q) ∧ r ⇐⇒ p ∧(q ∧ r), (p ∨ q) ∨ r ⇐⇒ p ∨ (q ∨ r).

p ∧ (q ∨ r) ⇐⇒ (p ∧ q) ∨ (p ∧ r),
p ∨ (q ∧ r) ⇐⇒ (p ∨ q) ∧ (p ∨ r).
∗
¬(p ∧ q) ⇐⇒ ¬p ∨ ¬q ¬(p ∨ q) ⇐⇒ ¬p ∧ ¬q.
p ⇒ q ⇐⇒ ¬p ∨q
p ⇒ q ⇐⇒ ¬q ⇒ ¬p
x
A x ∈ A
P A
x ∈ A
P A
x A P P (x)
“x ≤ 1” R x
0 “0 ≤ 1” x 2
“2 ≤ 1”

“x ≤ 1”
1
∗

§
P A x ∈ A P(x)
∀x ∈ A : P (x) ∀x[P(x)]
“ x ∈ X P (x)” ∀x[P (x)]
“x
2
≥ 0”
R ∀x[x
2
≥ 0]
P A A x
P (x)
∃x ∈ A : P (x) ∃x[P(x)]
“ x ∈ A P (x)” ∃x[P (x)]
x
2
− 3x + 2 = 0 R x = 1
∃x[x
2
− 3x + 2 = 0]
P A
¬(∀x[P (x)]) ⇐⇒ ∃x[¬P (x)]
¬(∃x[P (x)]) ⇐⇒ ∀x[¬P (x)]
¬(∀x[x
2
≥ 0]) ⇐⇒ ∃x[x

2
< 0]
¬(∃x[x
2
− 3x + 2 = 0]) ⇐⇒ ∀x[x
2
− 3x + 2 = 0]
P A B
x ∈ A y ∈ B P
P P(x, y)
x
0
∈ A y
0
∈ B
P (x
0
, y
0
)

P (x, y) A B
∃x[P (x, y)] ∀x[P (x, y)] B ∃y[P (x, y)]
∀y[P (x, y)] A
P (x, y) (∀x)(∀y)[P (x, y)] (∀x)(∃y)[P (x, y)]
(∃x)(∀y)[P (x, y)] (∃x)(∃y)[P (x, y)]
P A B
¬((∃x)(∀y)P (x, y)) ⇐⇒ (∀x)(∃y)(¬P (x, y))
¬((∃x)(∃y)P (x, y)) ⇐⇒ (∀x)(∀y)(¬P (x, y))
¬((∀x)(∀y)P (x, y)) ⇐⇒ (∃x)(∃y)(¬P (x, y))

¬((∀x)(∃y)P (x, y)) ⇐⇒ (∃x)(∀y)(¬P (x, y))
n
∃ ∀
“∀n ∈ N : P (n)”
P (n)
P
P

§
P
P
§
∗
A
A
⊲
⊲ x
2
− 3x + 2 = 0
⊲ N
⊲ Z 
∗

A B C
X Y Z
a b c x y z
1 2 x
2
−3x+ 2 = 0
1 2

x
2
− 3x + 2 = 0
x A x ∈ A x
A y A y /∈ A y
A
⊲ 5 ∈ N 2005 ∈ N −20 /∈ N
⊲ −20 ∈ Z
8
3
/∈ Z 
A B A
B A B A = B
A = B
x ∈ A ⇐⇒ x ∈ B.
A B
A = B A = B a ∈ A a /∈ B
b ∈ B b /∈ A
{ }
A = {a, b, c}
a b c 3 A “
A a b c”
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§
{1, 2, 3} = {3, 1, 2}
{1, 2, 3} = {1, 2, 3, 2}
⊲ {1, 2}
x
2
− 3x + 2 = 0 1 2

⊲
N = {0, 1, 2, 3, 4 . . .}
Z = {0, ±1, ±2, ±3, ±4, . . .}.

T
x A x T A
T A = {x : T (x)}
A = {x ∈ R : |x| ≤ 2}
−2 ≤ x ≤ 2 “A
|x| ≤ 2”
⊲
x
2
− 3x + 2 = 0

x ∈ R : x
2
− 3x + 2 = 0

.
⊲
Q =

m
n
: n ∈ N
+
, m ∈ Z

.



⊲

x ∈ Z : x
2
−3x
2
+ 2 = 0

= {1, 2}
⊲ {n ∈ N : 1 ≤ n ≤ 6} = {1, 2, 3, 4, 5, 6} 
∅
⊲ {x : x = x} = ∅
⊲ ∅ = {∅} {∅}
∅ 
A B A
B A B
A ⊆ B B ⊇ A A B
B A
A B A  B
A  B a ∈ A a /∈ B
⊲
N Z Q N ⊆ Z ⊆ Q
⊲ {a, {a}} {a} ∈ {a, {a}} {a} ⊆
{a, {a}}
⊲ A = {1, 2, 3, 4} B = {x ∈ N :
x } 1 ∈ A 1 /∈ B 

§

∅ ⊆ A A
A B C
A ⊆ A
A = B A ⊆ B B ⊆ A
A ⊆ B B ⊆ C A ⊆ C
C ∅ C = ∅
∅ ⊆ C 
A B A = B A
B A ⊂ B B
∅ B
A ⊂ B A B
b ∈ B b /∈ A
⊲ N ⊂ Z
⊲ A
1
= {x : 0 ≤ x ≤ 1} A
2
= {x : −1 ≤ x ≤ 2}
A
1
⊂ A
2
⊲ A = {1, 2, 3, 4} ∅ A
A A {1} {2} {3}
{4} {1, 2} {1, 3} {1, 4} {2, 3} {2, 4} {3, 4} {1, 2, 3} {1, 2, 4}
{1, 3, 4} {2, 3, 4} 

R
C Z
A A

(A) A
(A) A
(A) ∅ A (A)
A
{1, 2, 3, 4}
({1, 2, 3, 4}) = {∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3},{1, 4}, {2, 3},
{2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}.
A B A B
A B
A ∪ B A B
A ∪ B = {x : x ∈ A x ∈ B}.
A B A
B A ∩ B AB
A B
A ∩ B = {x : x ∈ A x ∈ B}.
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§
A ∩ B = ∅ A B A B
A = {a, b, c} B = {c, d}
A ∪ B = {a, b, c, d} A ∩B = {c}. 
A B A B
A B
A \ B
A \ B = {x : x ∈ A, x /∈ B}.
A = \B
B \ B = B
c
A \ B = A ∩ B
c
A = {a, b, c, d} B = {c, d, e} A\B = {a, b} 

A B C
A ∪ B = B ∪ A; A ∩ B = B ∩ A.
(A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C);
