Математика для бакалавров: универсальный курс для студентов гуманитарных направлений

П.В. Грес 

Математика для бакалавров 

Универсальный курс для студентов 
гуманитарных направлений 

Москва • 2020 • Логос 

УДК 51 (075.8) 
ББК 22.11я73 
 Г79 
Учебное пособие удостоено диплома 
Сибирского регионального конкурса 
«Университетская книга – 2010» 
в номинации «Лучший совместный проект» 

Рецензенты 
А.В. Пожидаев, доктор физико-математических наук, профессор 
Ю.И. Соловьев, доктор физико-математических наук, профессор 
П.Е. Алаев, доктор физико-математических наук  
А.К. Черненко, доктор физико-математических наук, профессор 

        Грес П.В. 
Г79    Математика для бакалавров. Универсальный курс для студентов гума
нитарных направлений: учеб. пособие / П.В. Грес. – Изд. 2-е, перераб. и доп. 
– М.: Логос, 2020. –   288 с.: ил. 

  ISBN 978-5-98704-751-4 

 Содержит краткий курс математики. Рассмотрены предмет математики, 
ее методологические проблемы и принципы, а  также  элементы теории 
множеств, дискретной математики и математической логики. Представлены важнейшие разделы математического анализа. Изложены математические методы, используемые в рамках теории вероятностей, математической статистики, математического моделирования и принятия решений. 
Приведены основные определения и методы, примеры решения типовых 
задач, задания для самостоятельной работы. В отличие от предыдущих  
изданий представлены разделы по линейной и векторной алгебре, аналитической геометрии, а также глубже рассмотрены вопросы теории вероятностей и математической статистики. В учебном пособии нашел отражение опыт преподавания  математики на гуманитарных специальностях  
вузов Новосибирска. Изложение материала адаптировано для обучения  
бакалавров. 
 Для студентов высших учебных заведений, обучающихся по направлениям и специальностям «Философия», «Психология», «Социология», 
«Юриспруденция», «Политология», «Социальная работа» и др. 

УДК 51 (075.8) 
ББК 22.11я73 

ISBN 978-5-98704-751-4    
© Грес П.В  ., 2020   
©  Логос, 2020 

Оглавление

Предисловие

ВВЕДЕНИЕ

1. Методологические проблемы математики

 

 ×π2. Теория множеств

∈∉⊂⊂⊂∅α∈α∈∈∪∪∈∈∪∩∩∈∈∩∩∅∈∉∆∆∪∆∩
∩∪∪∆∪∩∆ ∪∩∆∪ ∩ ∆∪∩ ∪ ∩ ∩ ∪∩ ∪= ∩∩= ∪∩ ∆= ∩ ∆∩ = ∩ = = = = = = α∈⊂≤≤ ≤≥≤
≤∈∈∈∈

∈∩∪∆∩∪∆∩∩∪∪≤ ≤ ≤ ≤ ∈∈3. Элементы дискретной математики

∑
=
=

1

∏
=
=

1

×≤≤=
−
−
−
=
[
][
]
[
]
−
=
+
−
−
−
−
−
−
=
1
0
0
0
0
=
= 13
12
11
10
9
8
7
6
5
4
3
2
1
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
13
16
3
16
16
3
16
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
=
=
−
=
=

== =
=
−
=
=
0

=

≤≤==
=
−
4
25
650
12
4
3
2
1
25
24
23
22
4
21
25
24
23
22
21
4
21
25
4
4
25
25
4
25
=
⋅
⋅
⋅
⋅
⋅
⋅
=
⋅
⋅
⋅
=
=
−
=

=4. Элементы математической логики

<
¬
¬⇒
⇒ ⇔
⇔ ∧
∧ ∨
∨ ∀
∀∃
∃∈¬∧∨⇒⇔⇒⇒¬¬¬¬⇒∨ ¬≡ ∧ ¬≡ ¬ ¬≡ ∨ ¬∧ ∨ ⇒ ⇔¬⇒ ¬¬  ¬ ⇒¬ ⇒¬ ∧∨∨∧∨∨ ¬∧ ∨ ∧ ∨∨ ¬∧ ⇔∨ ¬∨ ⇔¬∧ ¬¬∧ ⇔¬∨ ¬∨ ∧ ∨ ¬∨ ¬∧ ⇒ ∨ 5. Основы линейной алгебры

−
=
=
∆

2
12
10
3
4
5
2
5
4
3
2
−
=
−
=
⋅
−
⋅
=
=
∆

22
24
2
4
6
2
1
2
6
4
1
=
+
−
=
⋅
−
−
−
⋅
=
−
−
=
∆

2
1

4
3

4
7

2
5

5
2
3
1
−
−

5
3
10
6

3
2

2
1 −

−
−
−
+
+
=
=
∆

8
12
36
15
8
18
45
3
2
2
3
3
4

1
5
3
1
4
2
3
3
2
3
5
3
3
4
3
3
5
2
1
2
3

=
−
−
−
+
+
=
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∆

22222.
24
6
2
0
1
4
5
1
3
2

1
5
0
2
4
2
6
3
1

6
5
2

4
3
0

1
2
1

−
=
⋅
⋅
−
⋅
⋅
−
−
⋅
⋅
−
−

−
−
⋅
⋅
+
−
⋅
⋅
+
⋅
⋅
=

−

−

=
∆

○
○
○
○
○
○
○

○
○
○
○
○
○

○
○
○
○
○

1
5
2
0
1
1
4
3
1
2

3
4
0
2
1
2
5
1
1

5
4
2

1
1
0

3
2
1

−
=
⋅
⋅
−
⋅
−
⋅
−
⋅
⋅
−

−
⋅
⋅
+
⋅
−
⋅
+
⋅
⋅
=
−
=
∆

6
189
183
1
4
9
1
6
8
3
5
7

8
4
3
7
6
1
9
5
1
8
7
9
8
7
5
4
6
5
4
1
1
3
1
1

9
8
7
6
5
4
3
1
1

−
=
−
=
⋅
⋅
−
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
→
=
∆

1
5
2
0
1
1
4
3
1
2
3
4
0

2
1
2
5
1
1
4
2
5
4
2
1
0
1
1
0
2
1
3
2
1

5
4
2
1
1
0
3
2
1

−
=
⋅
⋅
−
⋅
−
⋅
−
⋅
⋅
−
⋅
⋅
+

+
⋅
−
⋅
+
⋅
⋅
=
−
→
−
=
∆

3
1
4
2
3
1
2
2
4
1
1
2

4
3
3
4
2
2

1
4
4
1
4

2
1
3
2
1

3
2
2
3
2

4
1
4

3
2
1

2
3
2

=
⋅
⋅
−
⋅
⋅
−
−
−
⋅
−
⋅
−
−
⋅
⋅
−
+

+
⋅
⋅
+
⋅
−
⋅
=

−
−

−
−

−

→

−

−

−

=
∆

4
3
2

1
2
1

2
1
1

2
3
1

6
5
4

3
2
1

−

1
2
2
2
1
2
2
2
1

−
−
−

1
4
3
1
3
2
0
2
2

−

−

3
2
1

6
3
5
0
1
2

−
−
−
−

2
0
2
3
1
2
1
2
3

−
−
−

1
2
3

4
5
6

7
8
9

1
2
1

0
1
1

3
2
2

−
−

=
+

=
+

2
2
2

1
1
1
2
2

1
1
=
∆

○
○
○
○
○

∆
∆
=
∆

∆
=

∆
∆
=
∆
∆
=
∆

∆
=

2
2

1
1
=
∆
2
2

1
1
=
∆

=
−
=
+
7
2
12
3
5
∆0
11
6
5
1
2
3
5
≠
−
=
−
−
=
−
=
∆

∆∆33
21
12
1
7
3
12
−
=
−
−
=
−
=
∆11
24
35
7
2
12
5
=
−
=
=
∆ 3
11
33 =
−
−
=
∆
∆
=
1
11
11
−
=
−
=
∆

∆
=






=
+
+
=
+
+
=
+
+

∆0
8
2
2
2
3
1
1

3
3
3
3
1
2
3
1
2
2
3
3

2
1
3

1
3
2

3
2
3

≠
−
=
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∆

8
2
2
8
0
1
1
3
3
0

3
1
8
0
1
2
2
3
0

2
1
0

1
3
8

3
2
0

−
=
⋅
⋅
−
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∆ 24
2
0
2
3
1
0
3
8
3

3
0
2
3
1
0
2
8
3

2
0
3

1
8
2

3
0
3

−
=
⋅
⋅
−
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∆ 24
0
2
2
3
8
1
0
3
3

0
1
2
3
8
2
0
3
3

0
1
3

8
3
2
0
2
3

=
⋅
⋅
−
⋅
⋅
−
⋅
⋅
−

−
⋅
⋅
+
⋅
⋅
+
⋅
⋅
=
=
∆ 1
8
8 =
−
−
=
∆
∆
=
3
8
24 =
−
−
=
∆

∆
=
3
8
24
−
=
−
=
∆
∆
=






=
−
+
−
=
+
−
=
+
+

1
3
4
2
2
2
2
2
3

0
15
4
18
4
6
16
3
1
3
4
2
1
2
1
2
3
≠
=
+
−
+
+
+
=
−
−
=
∆

15
4
12
1
6
4
2
1
3
1
2
1
2
1
2
2
−
=
−
−
+
−
+
=
−
−
−
=
∆30
4
6
8
2
16
6
1
1
4
2
2
2
1
2
3
=
+
−
+
+
+
=
−
−
=
∆ =
−
+
+
+
−
−
=
−
−
=
∆1
15
15
−
=
−
=
∆
∆
=
2
15
30 =
=
∆

∆
=
1
15
15 =
=
∆
∆
=
30
3000
100
3000
3000
+
=
+
+
=
+
100
==





=
+
+
=
+
+
=
+
+

125
3
4
115
2
5
130
4
3

0
40
3
1
4
2
5
1
1
4
3
≠
=
=
∆
800
3
1
125
2
5
115
1
4
130
=
=
∆600
3
125
4
2
115
1
1
130
3
=
=
∆ 400

125
1
4
115
5
1
130
4
3
=
=
∆ 20
40
800 =
=
∆
∆
=
15
40
600 =
=
∆

∆
=
10
40
400 =
=
∆
∆
=
0
=
∆

=
−
=
−
3
4
2
4
2
∆∆∆0
2
2
4
4
2
2
1
=
−
−
−
=
−
−
=
∆
10
6
16
4
3
2
4
−
=
+
−
=
−
−
=
∆5
8
3
3
2
4
1
−
=
−
=
=
∆ ∆ = 0∆≠ 0∆≠ 0
=
−
=
−
12
9
6

4
3
2
0
18
18
9
6
3
2
=
+
−
=
−
−
=
∆
0
36
36
9
12
3
4
=
+
−
=
−
−
=
∆0
24
24
12
6
4
2
=
−
=
=
∆ ∆ = 0∆= 0∆= 0
−
=
+
−
=
+
1
4
3
2
3

=
+
=
+
20
9
5
4
2






=
+
+
=
−
+
=
+
+

8
7
5
3
0
3
2
2








−
=
−
+
−
=
−
+
=
+
−

4
3
4

5
4
5
2
3
2







−
=
+
−
=
−
−
=
+
+

1
2
3
2
0
5
2
3
2







=
+
−
=
−
=
+

1
3
4
1
3
1
2







−
=
−
+
=
+
=
+
−

1
3
7
3
2
11
7







−
=
+
+
=
−
=
+
−

1
2
5
13
3
2

==========∅=========





=
+
+

=
+
+
=
+
+

1600
3
2
4
1000
2
3
2500
5
3
6

×

















=

2
1

2
22
21

1
12
11

=×














=
3
1
2
4
5
1
==





−
−
−
=
2
1
0
3
2
5
===
















−

1
4
2
3
1
0
0
1
1

====≠=(
)
2
1
=≠=

















2

1

===





=
6
5
4
3
2
1
=















6
3
5
2
4
1
















−
−
=
2
0
4
6
1
3
5
2
1
=(
)
3
0 −











−
−
=
2
6
5
0
1
2
4
3
1
=





− 3
0

=λ=λ==λλ






=
4
3
2
1
















λ
8
6
4
2
4
3
2
1
=====+





=
4
3
2
1






8
7
6
5






=






+
+
+
+
=
+
12
10
8
6
8
4
7
3
6
2
5
1









−
−
−
=








−
=
2
1
0
3
2
5
4
0
3
1
6
2






−
−
=
−






−
=
2
1
0
3
2
5
8
0
6
2
12
4
2






−
−
=






+
+
+
−
−
−
+
=
−
10
1
6
1
14
9
2
8
1
0
0
6
3
2
2
12
5
4
2













=
6
5
4
2
4
3
2
1
=+++











−
−
4
2
4
2
3
1






=






+
−
+
−
+
+
+
−
+
−
+
+
+
+
=
4
8
8
6
6
4
6
4
4
2
5
3
5
3
4
2
2
1
2
1







=






−
=
0
8
1
4
7
0
5
3
1
3
1
2
+














−
−
=
















−
−
=
0
4
5
2
1
0

5
1
2
3
1
6
=














=






−
=
0
3
3
0
1
1

4
1
0
5
4
1






5
5
2
7
8
2
















15
5
4
5
1
18

















8
25
13
8
5
7

=×=×==×∑
=
=

1







−
−
−
=






−
=
1
4
2
2
3
1
5
4
3
2
==






−
⋅
+
−
⋅
⋅
+
−
⋅
⋅
+
⋅

−
⋅
−
+
−
⋅
⋅
−
+
−
⋅
⋅
−
+
⋅
=
1
5
2
4
4
5
3
4
2
5
1
4

1
3
2
2
4
3
3
2
2
3
1
2






−
−
−
−
=






−
−
+
−
+
+
−
−
−
−
=
13
8
14
1
18
4
5
8
20
12
10
4
3
4
12
6
6
2

m 

n 

m 

p 
p 

A 
B 
C 

n 

• 
= 

• 
= 













=






=
5
3
1
4
2
1
1
1
1

6
5
4
3
2
1
=





=

=






⋅
+
⋅
+
⋅
⋅
+
⋅
+
⋅
⋅
+
⋅
+
⋅
⋅
+
⋅
+
⋅
⋅
+
⋅
+
⋅
⋅
+
⋅
+
⋅
=
⋅
=

54
32
15
24
14
6
5
6
4
5
1
4
3
6
2
5
1
4
1
6
1
5
1
4
5
3
4
2
1
1
3
3
2
2
1
1
1
3
1
2
1
1

(
)

















− 2
0
16
3
2
1






−
−






−
−
4
6
6
9
6
4
3
2






















2
2
1
2
2
1

1
1
3
0
1
2























6
5
4
3
2
1

6
5
4
3
2
1






















−
−

1
1
2

4
2
1
3
1
1






















6
5
4
3
2
1

6
5
4
3
2
1







0
0

0
0






9
7

5
4
















51
40
29
33
26
19
15
12
9









0

0 





64
49
28
22






=






−
−
=
1
3
1
1
2
1
1
3












+
−
+
−
−
−












−
−
⋅
1
5
2
0
2
1
6
1
1
3
3
3
1
3
1
1
2
1
1
3









−








+
−
−
+
−
−








−
−








⋅
1
8
1
2
2
3
1
9
2
1
1
3
2
1
1
3
1
3
1
1

≠





−
−
=
2
3
4
1
1
2















−
−
=
3
4
6
2
5
3















=
3
2
1
4
0
2
0
4
2















−
=
1
2
3
2
1
1
0
1
2


















−

−
−
−







−

1
1
3
1
0
4
2
5
1
0
1
2

2
1
0
4
5
3
1
2






−
−






−
1
4
3
2
1
4
2
1
































7
0
0
5
7
0
3
5
7

1
0
0
1
1
0
1
1
1
=





−
−
−
4
14
19
4
=















−
−
−
−

2
5
20
14
20
20
13
18
14

=














−

7
5
13
4
10
16
8
2
8

=











11
14
11
2
8
2
4
8
6

=





−1
6
2
0
2
24






−
−
4
12
8
4
















0
0
0
0
0
0
0
0
0
















−
=















=






=
1
1
2
1
1
0
0
1
1

3
1
2
1
1
1

6
5
4
3
2
1






=






+
+
+
+

+
+
+
+
=





















=
⋅
32
15

14
6
18
10
4
6
5
4
9
4
1
3
2
1

3
1

2
1
1
1

6
5
4
3
2
1
















=
















+
+
+
+
+
+
+
+
+
=






















=
⋅
21
17
13
15
12
9
9
7
5

18
3
15
2
12
1
12
3
10
2
8
1
6
3
5
2
4
1

6
5
4
3
2
1

3
1
2
1
1
1







=






+
+
+
+
−
+
+
+
+
+
+
−
+
+
=

=















−







=
⋅

11
7
16
5
4
7
6
5
0
6
5
4
12
0
4
3
2
0
3
2
1
6
0
1

1
1
2
1
1
0
0
1
1

6
5
4
3
2
1















−
−
=















−
















−
=
2
0
4
2
2
2
1
2
1

1
1
2
1
1
0
0
1
1

1
1
2
1
1
0
0
1
1
2

=+=+





=














 −
=



















=
1
4

1
1
2
2
1
1

4
3
3
3
2
2
2
1
1
1
0
0






































+
+
+
+
−
+
+
+
+
−
+
+
+
+
−
+
+
+
+















 −



















⋅

13
7
9
5
5
3
1
1

4
6
3
4
6
3
3
4
2
3
4
2
2
2
1
2
2
1

1
0
0
1
0
0

1
1
2
2
1
1

4
3
3

3
2
2
2
1
1
1
0
0



















=



















+
+
+
+

=

























=
⋅
⋅

41
29
17
5

13
28
9
20
5
12
1
4

1
4

13
7
9
5
5
3
1
1















 −
=
















+
+
+
−
=





















 −
=
⋅
5
10
3

1
4
2
8
1
4

1
4

1
1
2
2
1
1


















=



















+
+
−
+
+
−
+
+
−
+
+

=














 −



















=
⋅
⋅

41
29
17
5

20
30
9
15
20
6
10
10
3
5
0
0

5
10
3

4
3
3
3
2
2
2
1
1
1
0
0

= = (
)





 −

















−

−

1
15

5
2
2
4
3
1
3
2
1






















−
−



















1
4

1
1
2
2
1
1

4
3
3
3
2
2
2
1
1
1
0
0



















35
25
15
5

(
)
360
170
5
1
3
2
30
70
=







(
)
=
⋅
⋅
=








= = 

















=

1
0
0
0

0
1
0
0
0
1

≠=

1
=
= = 





=
−
=
+
=
+

10
5
16
3
5
2

0
29
1
5
0
3
0
1
0
1
2
≠
−
=
−
=
∆
=
















−
−
−
−
−
=
















−
=
∗

1
10
5
6
2
1
3
1
15

1
3
0
5
0
1
0
1
2
∗
=
1

















−
−
−
−
−
−
=
1
10
5
6
2
1
3
1
15

29
1















=
















−
−
−
−
=
















⋅
−
⋅
−
⋅
⋅
−
⋅
−
⋅
⋅
+
⋅
+
⋅
−
−
=

=
































−
−
−
−
−
−
=
=
−

5
3
1

145
87
29

29
1

10
1
16
10
5
5
10
6
16
2
5
1
10
3
16
1
5
15

29
1

10
16
5

1
10
5
6
2
1
3
1
15

29
1
1===





=
1
4

2
7















−
=
2
0
0
2
1
0
3
2
1







=
+
+
−
−
=
−
=
+
+

2
2
1

3
3
2
2








−
−
=
−
7
4
2
1
1















−
−
=
−

1
0
0
2
2
0
7
4
2

2
1
1
====
∆

−
−
−
=
∆
=
∆
=
∆





=
+
+
=
+
−
=
−
+


=
−
−
=
+






=
+
=
+
+
=
+






=
+
−
=
−
+
=
−
+

=+



=




=














−
−
=














−
−
=














−
=














=














=














−
−
=














−
−
=
6. Основы векторной алгебры

===λ ≠λ
λλ λλ=λ  =λ ⋅
+
⋅
−
=
⋅
−
=

 ⋅
−
=
+λ
+
λ
=
+

+

+
+
=
2
2
2
+
+
=
αβγ=
γ
=
β
=
α

α+β+γ=++++= λ λλλ =
=

0
5

6
3
2
1
≠
=
= 3
9
2
1
2
2
2
2
=
=
−
+
+
=
γ
β
α

====+=+=====2
5
3
−
+
=
3
−
=
=========
=
−
+
−
+
=
2
2
2
5
10
15
14
5
=
14
1
14
5
5
14
2
14
5
10
14
3
14
5
15
−
=
−
=
γ
−
=
−
=
β
=
=
α
=
= +−
2
α =α ==+3 74 74 74 74  ⋅
⋅
=

0
0
90
=
⋅
⋅
=
⋅
⋅
=
(
)
⋅
⋅

(
)++α(
)
(
)

−
=
⋅
−
=

=
−
+
⋅
+

−
⋅
+
⋅
=
=
=
α
2

1
−
=
α
α=π=++===++==
=
⋅
(
)
=
−
+
+

−
⋅
−
+
⋅
+
⋅
−
=
=

=++=++
=π=
⋅
⋅

××
−
−

×
189
8
2
11
2
2
2
=
+
+
−
=
⋅
×
=
⋅
⋅
×


=
3
1
2
2
2
2
2
=
+
−
+
=
7
6
3
2
2
2
2
=
+
+
=
10
10
15
6
3
2
1
2
2
+
−
−
=
−
=

17
5
425
10
10
15
2
2
2
=
=
+
−
+
−
×
982
0
21
17
5
7
3
17
5
≈
=
⋅
=
10
9
31
1
8
2
4
1
1
+
−
−
=
−
−
−
−
−
=

2
1142
10
9
31
2
1
2
1
2
2
2
=
+
−
+
−
=
×
=
=+=+2= = 
[
]
(
)
4
3
3
2
3
2
1
1
2

⋅
=
3
1
3
=
4
2
2
3
2
1
1
2
+
−
−
=
=

21
4
2
1
2
2
2
=
+
−
+
−
=
2
21
2
1
=
3
7

21
7

2
21
6
7
3
=
=
⋅
=
=

6
12
3

4
3
2
2
3
1

−
−
−
−

 = ++=

λ=++λ 3
3
4
=

⊥

7. Элементы аналитической геометрии

+
=
3
π
=
ϕ
=ϕ3 =3 ==3= π
=
ϕ
1
4 =
π
=
ϕ
=
= = += °= = +1
2

1

1
2

1
−
−
=
−
−
2
7
2
3
3
2
3
1
3
1
3
1
2
1
−
=
−
=
−
−
−
=
−
−
−
+= ++= = ++==−
−
=

−
=
++===−
=
=≠−
=
=≠−
=
========+==
−
−
=
−
+
−
=
+ −
=
−
=
====+=
−
=
−
=
−
=
−
=
=
−
+
−

++
+==+
+==
+ =
+
−
==+=
1
1
+
=
2
2
+
=
2
1

1
2
1+
−
=
ϕ
=0
1
2
1

1
2
=
+
−

=π∞
=
π
∞
=
+
−

2
1

1
2

1
+=1
2
1
−
=


+=+===+==−
=
−
−
+
−
−
−
=
ϕ

=+===
ϕ
ϕ ===−
=
−
=

3
1
2
1
=
= =++
−
=
=+==+=++=++=ϕ
2
1
2
1

1
2
2
1
+
−
=
ϕ
2

1

2

1
=
+=


=
+
+

=
+
+




+
=

+
=

2

1

2

1

2

1
2
1
≠
=
=
0
1
0
1
2
1
2
1
2
1
2
1
=
+
−
=
=
+
2
1
2
1

1
2
2
1

2
1

1
2

2

1

2

1
1
+
−
=
+
−
=
ϕ
≠
2

1

2

1

2

1
=
=

+=+=++
+=
=
−
+
=
+
−
==++====−
−
=
−
+=++=+=++=++=+=+=++=+==
−
α
+
α
αα

++=2
2
1

+
±
=
µ
+µ
µ+0

2
2
2
2
2
2
=
+
−
+
+
+


=
+
α
=
+
α
=
+
2
2
2
2
2
2

α+
+α=1

2
2

2
2

2
2

2

2
2

2
=
+

+
=
+
+
+
+=2
2
1

+
±
=
µ
=µ

+5
1

4
3

1

2
2
=
+
=
µ
µ=
−
+
=+=+=+=+==
−
+
−
√=
−
α
+
α
−
α
+
α
=
+
++=2
2

0
0

+

+
+
=
+=µ10
1

6
8

1

2
2
=
+
0
2
10
6

10
8
=
−
+
10
6
10
8
=
α
=
α
==
−
⋅
+
⋅
=

=
=
+

−
⋅
+
⋅
=
2
1
2
2
1
2
3
2
−
−
−
=
−
−

+=78
2
29
15
29
11
26

5
2

11
4
5
3
2

2
2
≈
=
−
=
+

−
⋅
+
⋅
=
+
+=+=+=5
5 2
2

2
1
0
2
1
0
+
=
+
=
2
2
3
7
3
2
3
9
0
0
=
−
=
−
=
+
−
=
7
3

7

12
2
12
−
−
−
=
−
−
=+=+9
3
9
7
3
7
−
−
−
−
=
−
−
−
2
1
6
5
−
−
=

6
5
−
5
6

6
5
1
=
−
−
=

5
18
5
6
7
5
6
12
+
=
−
=
−
1
2

1
2
−
−
=

2
9
4
2
6
3
5
3
3
2
0
3
7
6
4
3
6
0
=
−
−
−
=
−
=
−
−
−
−
=
=
−
−
−
=
75
0
4
3

7
6

2
9
7
6
2
9

=
=
⋅
+

−
=
+
−
=
∠=≈ =
⋅


 −
+

−
−
=
+
−
=

7
6
5
3
7
6
5
3

∠=≈=
⋅



−
+

−
−
=
+
−
=
∠=≈∠=∠∆ ∠4
3
4
6
0
6
−
−
−
=
−
−
+=4
2

4

6
3
6
−
−
=
−
−
−
=∠+
+
−
=
4
81
24
2
9
+
−
−
=−
−
−
=
+
−
=6
7
−
=
−
=
−
−
=
−
−
++=
=
−
−
=
+
+
0
2
3
5
0
4
6
7
=3
2
−
=




− 3
2
0++=+=++=+==+=++==2
2
−
+
−
2
2
0
2
0
=
−
+
−
==1

2

2

2

2
2
2
2
=
+
=
+
1

2

2

2

2
=
+
=ε=2
2
−
=
1

2

2

2

2
=
−
=

−
=
∞
→
2
2
2
2

2

2
2
−
±
=
−
=

∞
→
±
=
====2
1
4
1
2
2
2
=
⋅
=
=0
400
25
16
2
2
=
−
+

20
4
5
2
2
=
−
=2
2
2
4
2
=
+
+
ε=2
5
±
=
==0
0
0
0
−
−
−
=

=0
=++==++===++===++===+===+=======0
1
2
7
3
3
1
=
−
⋅
+
−
−
⋅
−
+
−
⋅
+=1
=
+
+
+
+=1
2
8
4
=
+
−
+
0
=
−
γ
+
β
+
α
αβγ2
2
2
1

+
+
±
=
µ
µ+=7
1
49
1

3
6
2

1

2
2
2
±
=
±
=
+
−
+
±
=
µ

7
1
=
µ
=7
1
=
µ
0
2
7
3
7
6
7
2
=
−
+
−
7
6

7
2
−
=
β
=
α

7
3
=
γ
=0

1
3
1
3
1
3

1
2
1
2
1
2

1
1
1
=
−
−
−
−
−
−
−
−
−

0

4
2
5
2
5
4
1
2
1
=
−
−
−
−
−
−
−
2
6
1
16
0
1
17
=
−
−
0
13
17
6
=
+
−
−