Джумадильдаев А.С. Элементы дискретной математики (часть 1)
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•
•
•
x ∈ M x M x ∈ M
x M
M x
1
,x
2
,...
M = {x
1
,x
2
,...}
M x P (x)
M = {x | P (x)}.
A
|A|.
|A|
|A| =42, A
N.
• N {1, 2, 3,...}
• Z
{...,−3, −2, −1, 0, 1, 2, 3,...}
• Z
+
{0, 1, 2,...}
• Q
{p/q : p, q ∈ Z,q =0}
• R
• C a + bi,
a, b ∈ R.
-
6
r
r
r
r
x
y
1
{(x, y) ∈ R
2
| 0 ≤ x ≤ 1, 0 ≤ y ≤ 1},
|x − 1/2|≤1/2, |y −1/2|≤1/2.
π/4
√
2
-
6
r
rr
r
@
@
@
@
x
y
1
x + y ≤ 1, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
x − y ≥−1,
−1 ≤ x ≤ 0, 0 ≤ y ≤ 1
x + y ≥−1,
−1 ≤ x ≤ 0, −1 ≤ y ≤ 1
x − y ≤ 1,
0 ≤ x ≤ 1, −1 ≤ y ≤ 0,
,
|x| + |y|≤1.
B
A ∀x ∈ B ⇒ x ∈ A.
∈ ⊂
1 ∈ N {1}⊂N. x
{x}
A
¯
A
∈ A ∈
¯
A.
A
∈ A.
∈
¯
A.
∈
¯
A.
∈ A.
∅
U
A B A = B A ⊆ B
B ⊆ A.
A = {a, b, c} B = {c, b, a, c}.
A ⊆ B,
a ∈ A ⇒ a ∈ B,
b ∈ A ⇒ b ∈ B,
c ∈ A ⇒ c ∈ B.
B ⊆ A,
c ∈ B ⇒ c ∈ A,
b ∈ B ⇒ b ∈ A,
a ∈ B ⇒ a ∈ A,
c ∈ B ⇒ c ∈ A.
A = B.
• A ∪ A = A, A ∩A = A
• A ∪ B = B ∪ A, A ∩ B = B ∩ A
• (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩B) ∩ C = A ∩ (B ∩ C)
• (A ∪ B) ∩ C =(A ∩ C) ∪ (B ∩ C), (A ∩ B) ∪ C =(A ∪ C) ∩ (B ∪ C)
• A ∪∅= A, A ∩ U = A,
A ∪ U = U, A ∩∅= ∅
•
¯
¯
A = A
• A ∪
¯
A = U, A ∩
¯
A = ∅
¯
U = ∅,
¯
∅ = U
• (A ∪ B)=
¯
A ∩
¯
B, (A ∩B)=
¯
A ∪
¯
B
X = Y
x ∈ X ⇒ x ∈ Y
y ∈ Y ⇒ y ∈ X.
A ∪ B =
&%
'$
(A ∪ B) ∪ C =
&%
'$
&%
'$
B ∪ C =
&%
'$
&%
'$
A ∪ (B ∪ C)=
&%
'$
&%
'$
(A ∪ B) ∪ C =
&%
'$
&%
'$
= A ∪ (B ∪ C)
(A ∪B)=
¯
A ∩
¯
B.
x ∈ (A ∪ B) ⇒ x ∈ U, x ∈ (A ∪ B) ⇒ x ∈ U, x ∈ A, x ∈ B ⇒ x ∈
¯
A, x ∈
¯
B.
(A ∪ B) ⊆
¯
A ∩
¯
B
x ∈
¯
A ∩
¯
B ⇒ x ∈
¯
A, x ∈
¯
B ⇒ x ∈ U, x ∈ A, x ∈ B ⇒ x ∈ U, x ∈ A ∪B ⇒ x ∈ (A ∪ B).
¯
A ∩
¯
B ⊆
(A ∪B).
N
•{5, 6}∈{{1, 2, 3, 5, 6}, {1, 3}, 3, 5, 6}
• 7 ∈{{5, 6, 7, 8}},
•{1}∈N
• 1 ∈ N
•{a}⊆{{a}}
•{a}∈{{a}}
• A ∈ B, B ∈ C, A ∈ C.
•
A ⊆ B, B ⊆ C A ⊆ C
•
x ∈ A {x}⊆A.
•
{x}⊆A, x ∈ A.
{a, b, c}, {c, b, a, c}, {b, c, b, a}, {c, a, c, b}
{{1, 2}} {1, 2}
{{1, 2}, {2, 3}} {1, 2, 3}