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10.0, 3, 7, 8, 11, 13*, 14, 15*.

11.f (x1, x2, x3) = x1x2 x3 .

Вариант 9

1. ((x1 x2) x3)x1 x2 x3 = (x2 x3) x1 x3 x1 .

2. f (x1, x2, x3)= (x1 x2)(x3 x1x2).

3. (

 

 

) (

 

 

x1)

 

 

 

 

 

x1

 

 

 

 

 

 

x2

x2

x3 = x2 x3 .

 

 

 

 

 

 

 

4. f (x1, x2, x3)= x1x2 x3 x1 .

5.1, 2, 4, 7, 8, 11, 13, 14.

6.1, 2, 4.

7.0, 1, 2, 3, 8, 9, 10, 11, 14, 15.

8.1, 2, 5, 7, 8, 12, 13, 14.

9.2, 4, 7, 9, 10, 14, 15.

10.0, 1, 2, 3, 8*, 9, 10, 11*, 14, 15*.

11.f (x1, x2, x3)= x1x2 x3 .

Вариант 10

1.(x1 x2)(x3 x1) 1 → x3x4 = (x1x2 (x3 x1))(x3 x4).

2.

f (

 

1,

 

2,

 

3)= (

 

 

2

 

 

 

 

4 (

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

3)

 

1

 

2) .

 

 

x

 

x

 

x

 

x

x

 

x

 

x

 

x

 

x

 

 

 

 

 

 

 

 

 

 

 

 

3. x1x2 x3 x1x3 (x1 x2)= (x1 x2) x3 (x1 x2)(x1 x2).

4.f (x1, x2, x3)= x1x2 x3 x1x3 .

5.3, 7, 11, 15.

6.0, 1, 2, 3, 4, 5, 6.

7.4, 6, 7, 8, 9, 10, 14, 15.

8.1, 5, 6, 7, 8, 9, 10, 15.

9.0, 3, 7, 8, 9, 10, 11, 12, 15.

10.1, 5*, 6, 7, 8, 9*, 10, 15.

11.f (x1, x2, x3)= x1x2 x3 .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Вариант 11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.(x1x2

x3)x4

(x1

x2)

= (x1x2

x3)x4

(x1 x2)x4

(x1

x2) .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. f (x1, x2, x3)= (x1 x2)(x3 x2 x1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

(x1x2

x3)x1

x2

=

(x1

x2)&x3

x1

 

x2 .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4. f

(x1, x2, x3)= x1

 

 

 

.

 

 

 

 

 

 

x3

x2

 

 

 

 

 

 

5.1, 2, 3, 4, 5, 6, 7, 9, 13, 14, 15.

6.1, 2, 3, 4, 5, 6.

21

7.0, 1, 2, 3, 4, 8, 9, 10 , 11, 12.

8.0, 1, 2, 3, 9, 10, 13, 14, 15.

9.0, 2, 3, 5, 11, 12, 15.

10.0, 2*, 3, 5, 11, 12*, 15*.

11.f (x1, x2, x3) = x1 x3 x2 .

Вариант 12

1.(x1x2 x3) (x1x3 x2) (x1x2 x3)= x1 (x2 x3).

2.f (x1, x2, x3)= (x1x2 x3 x1) (x1 x2 x3).

3.(x1 x2)(x1 x2)(x1 x2)(x1 x2) = (x1 x2)(x1 x2)(x1 x2).

4.f (x1, x2, x3)= (x1 x2) (x2 x3).

5.0, 3, 4, 7, 8, 11, 12, 15.

6.2, 3, 4, 5.

7.1, 3, 6, 8, 9, 10, 12, 13.

8.0, 2, 5, 6, 8, 11, 12, 13, 14.

9.0, 2, 4, 7, 8, 10, 13, 15.

10.0, 2, 5, 6*, 8, 11*, 12*, 13, 14.

11.f (x1, x2, x3)= (x1x2) (x2 x3).

Вариант 13

1.(x1 x2)((x1 x2)(x1 x2))= x1 x2 (x1 x2).

2.f (x1, x2, x3)= ((x1 x2) x1 (x3 x2)) x3 .

3.((x1 x2) x3)x1 x2 x3 = (x2 x3) x1 x3 x1 .

4.f (x1, x2, x3)= x3x1 x2 x1 .

5.0, 1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15.

6.0, 2, 3, 4, 5, 7.

7.2, 5, 7, 8, 11, 12, 13, 15.

8.1, 4, 6, 7, 10, 11, 12, 14.

9.0, 3, 6, 8, 9, 12, 13, 15.

10.0, 3, 6, 8, 9*, 12, 13*, 15*.

11.f (x1, x2, x3)= x3x1 x2 x1 .

Вариант 14

1.(x1x2 x3 x1) x1 x2 x3 = x3 (x1 x2) x3((x1 x2)(x1 x2 )) x1 .

2.f (x1, x2, x3)= x1 x2 x3 x1x2 .

3.(x2 x3 x1x2)x3 x2 x1 = (x2 x3 x1x2) x3 x2 x1 .

4.f (x1, x2, x3)= x3 (x1 x2 x3) x2 .

5.0, 1, 6, 7, 8, 9, 14, 15.

6.0, 3, 4, 6, 7.

22

7.1, 3, 6, 7, 9, 12, 13, 14, 15.

8.0, 2, 5, 7, 8, 9, 11, 12.

9.2, 4, 7, 9, 10, 11, 13, 15.

10.0*, 2, 5, 7, 8, 9*, 11*, 12.

11.f (x1, x2, x3) = x3 (x1 x2 x3).

Вариант 15

1. (x1 x2) ~ ((x1 ~ x2) (x1 | x2)) = x2 x1 .

2. f (x1x2x3) = x1 | (x3x2) .

3. x1 x2 ~ x3 (x1 x2 ) = x1x2x3 x3x1x2 (x1 ~ x2 ) .

4.f (x1, x2 , x3, x4 ) = x1|x2 (x3 x4 ) .

5.2, 3, 4, 5, 8, 9, 14, 15.

6.0, 1, 2, 5.

7.0, 1, 6, 7, 8, 9, 10, 11.

8.3, 5, 6, 7, 8, 9, 13, 14.

9.5, 7, 8, 9, 10, 11, 15.

10.0*, 1*, 6*, 7, 8, 9, 10, 11.

11.f (x1, x2 , x3) = x1|x2 x3 .

Вариант 16

1.(x1 x2) (x2 x1)x1 = (x1 x2) 1(x2 x1 x2 x1x2).

2.f (x1, x2, x3)= x1 x2 (x1 x2) x3 .

3.(x1x2 x3) (x1 x2 x3) (x1 x2) x3 = (x1 x2)(x2 x3)x3 .

4.f (x1, x2, x3)= (x1 x2) (x2 x3).

5.0, 1, 2, 3, 12, 13, 14, 15.

6.1, 2, 3, 7.

7.0, 1, 2, 3, 4, 10, 11, 14, 15.

8.0, 1, 2, 3, 7, 8, 9, 10, 11, 15.

9.2, 3, 4, 5, 9, 10, 12, 13.

10.0, 1*, 2*, 3, 7*, 8, 9, 10, 11, 15.

11.f (x1, x2, x3)= (x1 x2).

Вариант 17

1. ((x1 x2 ) ((x1 x2 ) (x1 ~ x2 ))) = x1 x2 . 2. f (x1x2x3) = x1(x2|x1) .

23

3.[(x1 x2) x1] x2 x1 = x2x1 .

4.f (x1, x2, x3) = (x1~x2)&x3 .

5.1, 2, 4, 7, 9, 10, 13, 15.

6.1, 2, 4, 5.

7.2, 3, 6, 8, 9, 13, 14, 15.

8.0, 1, 2, 7, 8, 9, 10, 13, 14, 15.

9.2, 3, 4, 9, 10, 11, 14, 15.

10.0, 1*, 2, 7*, 8*, 9*, 10, 13, 14, 15.

11.f (x1, x2, x3) = (x1 x2)&x3 .

Вариант 18

1. ([(x1 x2 ) x3] [x1 | (x2 | x3)]) 1 = (x1 x2 | x3 ) ~ (x1 x2 x3) .

2. f (x1, x2 , x3 ) = (x1 x2) [(x2 x3) x3] .

3.(x1x3 x2)(x3 x1) ~ x2 = x1 | (x3x2) .

4.f (x1, x2 , x3) = (x1 x2 x3) x3 .

5.4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15.

6.0, 3, 4, 7.

7.3, 4, 5, 6, 13, 14, 15.

8.2, 6, 7, 10 ,12 ,13 14, 15.

9.0, 2, 4, 8, 12, 14, 15.

10.2*, 6, 7*, 10 ,12 ,13 14, 15.

11.f (x1, x2 , x3) = (x1 x2 x3) .

Вариант 19

1. (x1 → x2 ) (x1 x2 ) (x1 x2 ) = (x1↓x2 ) → x1x2 .

2.f (x1, x2 , x3) = (x2 ~ x3) x1→x3 x1 .

3.(x1x2 x3) (x4~(x1x2)) = (x1x2x3) (x4 (x1x2)) (x4 (x1 x2)) .

4.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .

5.0, 10, 11, 12, 13, 14, 15.

6.1, 2, 5, 6.

7.7, 8, 9, 11, 12, 14.

8.1, 2, 3, 4, 9, 11, 12, 14.

9.3, 4, 5, 6, 11, 13, 15.

10.3*, 4*, 5, 6*, 11, 13, 15.

11.f (x1, x2, x3) = x1x2 x1x2x3 .

24

Вариант 20

1.[(x1 x2) x1]x2x1 = x2 x1 .

2.f (x1, x2 , x3) = (x1 x2 | x3) ~ (x1 x2 x3) .

3.(x1x2)[(x3 x1) 1]x3x4 = (x1x2 (x3 ~ x1))(x1 | x4) .

4.f (x1, x2 , x3) = x1x2 x1x3 x2x3 .

5.0, 2, 4, 6, 9, 11, 13, 15.

6.0, 1, 2, 3, 5, 6, 7.

7.3, 5, 6, 7, 9, 12, 13, 15.

8.3, 4, 5, 6, 7, 8, 9, 10, 11.

9.5, 6, 7, 8, 9, 10, 11, 12, 13.

10.5*, 6, 7*, 8*, 9, 10*, 11, 12, 13.

11.f (x1, x2 , x3) = x1x2 x1x3 .

Вариант 21

1.(x1 ~ x2) [(x2 x1) x1] = (x1 x2) 1 (x1x2 x2 x1x2) .

2.f (x1, x2, x3) = x1x2x3 x3x1x2 .

3.(x1 x2 ) (x1 x2 ) (x1 x2) = (x1x2) x1x2 .

4.f (x1, x2, x3) = [(x2 x3 x1x2) | x3] | x2x1 .

5.0, 1, 2, 3, 7, 11, 15.

6.0, 1, 4, 5.

7.2, 7, 8, 9, 10, 11, 12, 13, 14.

8.0, 1, 5, 6, 8, 11, 12.

9.2, 3, 7, 8, 10, 13, 14, 15.

10.2*, 3, 7*, 8*, 10, 13*, 14, 15.

11.f (x1, x2, x3) = x1x2x3 .

Вариант 22

1.(x1x2x3) (x1 x2x3) (x1 x2) x3 = (x1 x2) [(x2 x3) x3] .

2.f (x1, x2, x3) = (x1x2) (x2x1) x3 .

3.[x1x2x3 (x3x1 x1x2x3)] 1 = [x1 → (x2 ↓ x3)] ~ [(x1 x3)] ↓ x1x2] .

4.f (x1, x2 , x3) = (x1 x2)(x1 x2 )x3 .

5.2, 3, 4, 5, 10, 11, 12, 13.

6.1, 2, 4, 5, 6.

7.3, 4, 6, 7, 8, 9, 11, 13, 15.

25

8.1, 2, 4, 6, 7, 9, 10, 14, 15.

9.2, 3, 4, 6, 8, 9, 11, 12.

10.3*, 4*, 6, 7*, 8, 9*, 11, 13, 15.

11.f (x1, x2 , x3) = (x1 x2)(x1 x2 ) .

Вариант 23

1.f (x1, x2 , x3) = [(x1x2x3)x1] x2 .

2.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .

3.(x1 ~ x2 ) [(x2 → x1) ↓ x1] = (x1 x2) 1 (x1x2 x2 x1x2) .

4.f (x1, x2, x3) = [(x1x2x3)x1] x2 .

5.0, 2, 5, 7, 8, 10, 13, 15.

6.1, 2, 3, 5, 6, 7.

7.7, 9, 10, 13, 14, 15.

8.0, 1, 2, 6, 10, 12, 13, 14.

9.2, 3, 4, 8, 12, 14, 15.

10.2*, 3, 4*, 8*, 12*, 14, 15.

11.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .

Вариант 24

1. (x1x2 x3) | (x1 x2 x3) = (x1x2|x3) (x2 x1x3) .

2.f (x1, x2, x3) = [(x2 x3 x1x2) | x3] | x2x1 .

3.(xy z) (xz | y) (xy z) = x(y|z) .

4.f (x1, x2 , x3 ) = x1x2x3 x3x1x2 .

5.0, 1 , 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15.

6.0, 2, 3, 4, 6, 7.

7.4, 7, 8, 9, 11, 12, 15.

8.1, 2, 4, 5, 9, 10, 13, 14.

9.0, 3, 4, 6, 7, 11, 12, 15.

10.4*, 7*, 8, 9, 11*, 12, 15.

11.f (x1, x2 , x3 ) = x1x2x3 .

Вариант 25

1.[(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .

2.f (x1, x2, x3) = (x1x2 x3) [(x2~x1) x3] .

3.(x1x2x3x1) (x1 ~ x2 ~ x3) = x3(x1 ~ x2) x3((x1x2)(x1x2)) x1x2 x3 .

26

4.f (x1, x2, x3) = (x1 x2)(x1 x2) ~ x3 .

5.0, 4, 8, 11, 12, 13, 14, 15.

6.0, 1, 3, 4, 6, 7.

7.6, 7, 8, 9, 10, 14, 15.

8.1, 3, 5, 8, 9, 10, 11, 12.

9.3, 5, 7, 10, 11, 12, 13, 14.

10.1*, 3, 5*, 8, 9, 10, 11, 12.

11.f (x1, x2, x3) = (x1 x2)(x1 x3) .

Вариант 26

1.(x1 x2 ) (x1 x2 ) (x1 x2) = (x1x2) x1x2 .

2.f (x1, x2 , x3) = [(x1x2x3)x1] x2 .

3.(x1x2 x3) (x4~(x1x2)) = (x1x2x3) (x4 (x1x2)) (x4 (x1 x2)) .

4.f (x1, x2 , x3 ) = x1x2 x3 x1x2x3 .

5.0, 10, 11, 12, 13, 14, 15.

6.1, 2, 5, 6.

7.7, 8, 9, 11, 12, 14.

8.1, 2, 3, 4, 9, 11, 12, 14.

9.3, 4, 5, 6, 11, 13, 15.

10.1*, 2, 3*, 4, 9*, 11, 12, 14.

11.f (x1, x2 , x3 ) = x1x2 x3 x1x2x3 .

Вариант 27

1. [(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .

2.f (x1, x2 , x3) = (x2 ~ x3) x1→x3 x1 .

3.(x1x2x3) (x1 x2x3) (x1 x2) x3 = (x1 x2) [(x2 x3) x3] .

4.f (x1, x2, x3)= (x1 x2)(x3 x1x2).

5.0, 4,8,9,10, 11, 12, 13, 14.

6.1, 3, 5, 7.

7.7, 8, 9, 10,11, 12, 14.

8.0, 2, 3, 4, 10, 11, 12, 15.

9.3, 4, 5, 9, 11, 13, 15.

10.0*, 2*, 3, 4*, 10, 11, 12, 15.

11.f (x1, x2 , x3) = (x2 ~ x3) x1x3 .

Вариант 28

1. (x1x2 x3) (x4~(x1x2)) = (x1x2x3) (x4 (x1x2)) (x4 (x1 x2)) .

27

2. f (x1, x2 , x3) = (x2 ~ x3) x1x3 x1 .

3. [(x1 x2 ) (x1 ~ (x1 ~ x2))] 1 = x1 | x2 .

4.f (x1, x2 , x3) = [(x1x2x3)x1] x2 .

5.3,4, 10, 11, 12, 13, 14, 15.

6.1, 2,4,5, 6.

7.1,3,5,7, 8, 9, 11, 12, 14.

8.1, 2, 3, 4,7,8, 9, 11, 12, 14.

9.1,2,3, 4, 5, 6, 11, 13, 15.

10.1*, 2, 3*, 4*,7,8, 9*, 11, 12, 14.

11.f (x1, x2, x3)= (x1 x2)(x3 x1x2).

Вариант 29

1.(x1x2x3) (x1 x2x3) (x1 x2) x3 = (x1 x2) [(x2 x3) x3] .

2.f (x1, x2, x3)= (x1 x2)(x3 x1x2).

3. ((x1 x2 ) ((x1 x2) (x1 ~ x2))) = x1 x2 .

4.f (x1, x2 , x3) = x1x2 x3 x1x2x3 .

5.1,3 , 4, 10, 12, 14, 15.

6.0, 3,4, 5, 6.

7.7, 8, 9, 10, 12, 15.

8.1, 2, 3, 4, 5, 11, 13, 14.

9.2, 4, 5, 6, 7, 10, 15.

10.1*, 2*, 3, 4*, 5, 11*, 13, 14.

11.f (x1, x2, x3)= (x1x2)(x3 x1x2).

 

 

 

 

 

 

 

Вариант 30

 

 

 

 

 

 

 

 

 

 

1.

((x1 x2 ) ((x1 x2 ) (x1 ~ x2 ))) = x1 x2 .

2.

f (x1, x2, x3)= (x1x2) (x3

 

 

 

).

x1

x2

3.(x1x2 x3) (x4~(x1x2)) = (x1x2x3) (x4 (x1x2)) (x4 (x1 x2)) .

4.f (x1, x2, x3)= (x1 x2)x3 (x1 x2).

5.0, 1,10, 11, 12, 13, 14, 15.

6.3, 4, 5, 6.

7.7, 9, 11, 12, 14.

8.2, 3, 7, 9, 11, 12, 14.

9.2, 4, 5, 8, 11, 13, 15.

10.2*, 3*, 7, 9*, 11, 12*, 14.

11.f (x1, x2, x3)= (x1 x2)x3 .

28

3. СБОРНИК ЗАДАНИЙ ДЛЯ СЕМЕСТРОВОЙ РАБОТЫ № 2

ПО КУРСУ «Дискретная математика»

3.1. Пример решения и оформления

Задание

Дано:

Х2 V4 Х3

V2

V3

V5

V6

X1 V1 X4 V7 X5

Задание 1.

Задать граф следующими способами: перечислением, матрицами смежности и инцидентности.

Решение:

 

 

 

 

 

 

 

 

 

Перечисление:

 

 

 

 

 

 

 

 

 

Множество вершин: X =

x1,x2,x3,x4,x5 .

 

 

 

Множество связей:

V =

 

<x4 x1>, <x2 x1>, <x2 x4>, <x2 x3>,<x4 x3>

 

 

 

 

 

<x5x3>, <x4 x5>

 

 

 

Множество изолированных вершин: пусто.

 

 

 

Матрица инцидентности:

 

 

 

 

 

 

 

 

 

 

V1

V2

 

V3

V4

V5

V6

V7

 

 

X1

 

-1

-1

 

0

0

0

0

0

 

 

X2

 

0

1

 

1

1

0

0

0

 

 

X3

 

0

0

 

0

-1

-1

-1

0

 

 

X4

 

1

0

 

-1

0

1

0

1

 

 

X5

 

0

0

 

0

0

0

1

-1

 

Матрица смежности:

 

X1

X2

X3

X4

X5

X1

0

0

0

0

0

X2

1

0

1

1

0

X3

0

0

0

0

0

X4

1

0

1

0

1

X5

0

0

1

0

0

29

Задание 2.

Определить следующие основные характеристики графа: число ребер и дуг; число вершин; коэффициент связности графа; степени всех вершин; цикломатическое число графа.

Решение:

Число ребер – 0 ; дуг – 7. Число вершин – 5.

Коэффициент связности графа - 1.

Степени всех вершин:

 

Х1

Х2

Х3

Х4

Х5

Полустепень исхода

0

3

0

3

1

Полустепень захода

2

0

3

1

1

Степень

2

3

3

4

2

Цикломатическое число графа = (число связей – число вершин) + коэффициент связности. Т.е. 7- 5+1 = 3; цикломатическое число равно 3.

Задание 3.

Определить, является ли данный граф:

-планарным или плоским графом (обосновать ответ и выполнить обратное преобразование);

-двудольным графом (обосновать ответ и, если необходимо, то достроить до двудольного графа);

-деревом (обосновать ответ и, в случае циклического графа, привести один из вариантов основного дерева);

-псевдографом или мультиграфом, или простым графом (обосновать ответ и выполнить необходимые преобразования).

Решение:

Данный граф является плоским, поскольку все его связи пересекаются только в вершинах. Преобразуем данный граф в планарный граф:

 

Х2

V4

Х3

 

V2

V3

V5

 

V6

X1

V1

X4

V7

X5

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