шамин с сдо
.pdfA B
A B
A
B
{1, 2, 3, 4} {2, 3, 4, 5} = {1, 2, 3, 4, 5}.
A B
A ∩ B
AB
AB
A ∩ B = .
A ∩ B = B ∩ A, A B = B A.
A \ B
AB
A B
A \ A = .
A B
A B = (A \ B) (B \ A).
U
A A U
A
A = U \ A.
CA
A = [0, 1], B = (0, 1), C = [0, 2].
A ∩ B = {0, 1}, A C = A, C \ A = (1, 2].
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f : A → B |
y = f(x) |
x A y B |
f
A B
f |
A |
B |
f : [0, 1] → [0, 1],
f(x) = x2.
f(x)
sign(x) = |
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0, |
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x = 0, |
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1, |
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x > 0, |
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1, |
x < 0. |
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sign(x) |
− |
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{−1, 0, 1} |
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1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
−10.0 |
−7.5 |
−5.0 |
−2.5 |
0.0 |
2.5 |
5.0 |
7.5 |
10.0 |
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x |
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f(x) = x2
f(x)
y
x |
f(x) = y |
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f(x) |
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x = √y [0, 1] |
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y |
[0, 1] |
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f(x) = y |
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1.0 |
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0.8 |
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) |
0.6 |
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f(x |
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0.4 |
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0.2 |
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0.0 |
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0.0 |
0.2 |
0.4 |
x |
0.6 |
0.8 |
1.0 |
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f(x) = x2 |
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f(x) = x2
[−1, 1]
N
N = {1, 2, 3, . . . }.
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N |
Z |
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Q |
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a = |
p |
, |
p, q Z, q 6= 0. |
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q |
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R
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a |
b |
a < b, a = b |
a > b. |
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a < b |
b < c |
a < c |
a R |
n Z |
n > a.
[ak, bk] |
k = 1, 2, . . . |
[ak+1, bk+1] [ak, bk], k = 1, 2, . . . .
1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
−1.00
π
a1 |
a2 |
a3 a4 a5 |
b5 b4 b3 |
b2 |
b |
1 |
2.25 |
2.50 |
2.75 |
3.00 x 3.25 |
3.50 |
3.75 |
4.00 |
c [ak, bk], k = 1, 2, . . . .
A |
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A R |
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A |
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b R |
x A |
x ≤ b, |
(b ≤ x). |
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A |
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a |
A |
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x A |
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x ≤ b, (b ≤ x).
(0, 1)
a (0, 1)
(0, 1)
b = (1 + a) 2
(0, 1) |
b > a |
a
c = a2 (0, 1),
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b |
A |
x A |
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x ≤ b |
b′ < b |
x0 A |
x0 > b′.
b = sup A
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b |
A |
x A |
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x ≥ b |
b′ > b |
x0 A |
x0 < b′.
b = inf A
A = (0, 1)
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inf A = 0, sup A = 1. |
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b′ > 0 |
x (0, 1) |
x ≥ 0 |
x0 < b′ |
x0 (0, 1) |
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x0 = b′/2 |
b′ ≤ 1 |
b′ > 1 |
x0 |
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(0, 1) |
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A |
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