Математика для гуманитариев: общий курс

Рассмотрены предмет математики и ее 
методологические проблемы и принципы. Даны элементы теории множеств 
дискретной математики и математической логики, линейной и векторной 
алгебры. 
Представлены 
элементы 
аналитической геометрии и важнейшие разделы математического анализа. Изложены математические методы 
теории вероятности, математической 
статистики, математического моделирования и теории принятия решений.

Приведены определения, примеры решения типовых задач, задания для самостоятельной работы.

ISBN 978-5-98704-297-6

ÓÄÊ 51(075.8) 
ÁÁÊ 22.11ÿ73 
Ã79 

Ñåðèÿ îñíîâàíà â 2003 ãîäó 

Ðåöåíçåíòû 
À.Â.  Ïîæèäàåâ,  äîêòîð  ôèçèêî-ìàòåìàòè÷åñêèõ  íàóê,  ïðîôåññîð 
Þ.È.  Ñîëîâüåâ,  äîêòîð  ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,  ïðîôåññîð 
Ï.Å.  Àëàåâ,  äîêòîð  ôèçèêî-ìàòåìàòè÷åñêèõ  íàóê 
À.Ê.  ×åðíåíêî,  äîêòîð  ôèçèêî-ìàòåìàòè÷åñêèõ  íàóê,  ïðîôåññîð 

Ã79  

Ãðåñ Ï.Â. 
Ìàòåìàòèêà  äëÿ  ãóìàíèòàðèåâ. Îáùèé  êóðñ:  ó÷åá.  ïîîáèå. — 
Èçä. 2-å,  ïåðåðàá.  è äîï. / Ï.Â.  Ãðåñ — Ì.: Ëîãîñ,  2020. — 288 ñ.: 
èë. — (Íîâàÿ  óíèâåðñèòåòñêàÿ áèáëèîòåêà). 
ISBN  987-5-98704-785-9 

Ñîäåðæèò  êðàòêèé  êóðñ ìàòåìàòèêè.  Ðàññìîòðåíû ïðåäìåò  ìàòå- 
ìàòèêè,  åå ìåòîäîëîãè÷åñêèå ïðîáëåìû  è ïðèíöèïû,  à òàêæå  ýëå- 
ìåíòû  òåîðèè  ìíîæåñòâ,  äèñêðåòíîé  ìàòåìàòèêè  è ìàòåìàòè÷åñ- 
êîé ëîãèêè. Ïðåäñòàâëåíû âàæíåéøèå ðàçäåëû ìàòåìàòè÷åñêîãî 
àíàëèçà.  Èçëîæåíû  ìàòåìàòè÷åñêèå ìåòîäû,  èñïîëüçóåìûå â ðàì- 
êàõ òåîðèè  âåðîÿòíîñòåé, ìàòåìàòè÷åñêîé ñòàòèñòèêè,  ìàòåìàòè- 
÷åñêîãî  ìîäåëèðîâàíèÿ è ïðèíÿòèÿ  ðåøåíèé.  Ïðèâåäåíû  îñíîâ- 
íûå îïðåäåëåíèÿ  è ìåòîäû, ïðèìåðû  ðåøåíèÿ  òèïîâûõ  çàäà÷, çàäà- 
íèÿ äëÿ ñàìîñòîÿòåëüíîé ðàáîòû.   îòëè÷èå  îò ïðåäûäóùåãî èçäà- 
íèÿ (Ì.: Ëîãîñ, 2003) ïðåäñòàâëåíû ðàçäåëû ïî ëèíåéíîé  è âåêòîð- 
íîé àëãåáðå,  àíàëèòè÷åñêîé ãåîìåòðèè,  à òàêæå  ãëóáæå  ðàññìîòðå- 
íû âîïðîñû  òåîðèè  âåðîÿòíîñòåé è ìàòåìàòè÷åñêîé ñòàòèñòèêè. 
 ó÷åáíîì  ïîñîáèè  íàøåë  îòðàæåíèå  îïûò ïðåïîäàâàíèÿ ìàòåìà- 
òèêè íà ãóìàíèòàðíûõ ñïåöèàëüíîñòÿõ âóçîâ  Íîâîñèáèðñêà. 
Äëÿ ñòóäåíòîâ  âûñøèõ  ó÷åáíûõ  çàâåäåíèé,  îáó÷àþùèõñÿ ïî íà- 
ïðàâëåíèÿì  è ñïåöèàëüíîñòÿì «Ôèëîñîôèÿ», «Ïñèõîëîãèÿ», «Ñî- 
öèîëîãèÿ», «Þðèñïðóäåíöèÿ», «Ïîëèòîëîãèÿ», «Ñîöèàëüíàÿ ðàáîòà» 
è äð. 
ÓÄÊ 51(075.8) 
ÁÁÊ 22.11ÿ73 

ISBN  978-5-98704-785-9
© Ãðåñ Ï.Â., 2020 

© Ëîãîñ, 2020 

Оглавление

Предисловие

ВВЕДЕНИЕ


1. Методологические проблемы математики




×π 

2. Теория множеств

∈∉⊂⊂⊂∅α∈α∈∈∪∪∈∈∪∩∩∈∈∩∩∅∈∉∆∆∪∆∩∩∪∪∆∪∩∆ ∪∩∆∪ ∩ ∆∪∩ ∪ ∩ ∩ ∪∩ ∪= ∩∩= ∪∩ ∆= ∩ ∆∩ = ∩ = = = = = = α∈⊂≤≤ ≤≥≤≤ ∈∈∈∈∈∩∪∆∩∪∆∩∩∪∪≤ ≤ ≤ ≤ ∈∈3. Элементы дискретной математики

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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
=
+
−
−

+++=+++===(
)

+
+
⋅
+
+

+
+
=

=
=
ϕ

=
=
=
+
+
+=++=== =
−
+
+
⋅
+
−
+

−
⋅
+
⋅
−
+
⋅
=
ϕ

++=+++====++=++=


=
+
+
+

=
+
+
+

= −
−
−
=

= 
−
=
−
=
−

{
}−
−
−
−
−
=
−
−
=
−
−

10
2
6
7
4
1
−
=
−
−
=
−

5
2
3
7
2
1
−
=
−
−
=
−

+=+=+=++=+=+=+=+=+=+=+=+=8. Введение в математический анализ



∈∈=∈∈∈=∈→εδδ≠ε
→=
→
→
==



 +
∞
→

=+ε

ε

δ
δ