Дискретка.Лекции, литература / Chast1_2012

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9.Проверить эквивалентность формул и , построив таблицы.

9.1.= x1(x2 _ x3), = x1x2 _ x1x3

9.2.= x1(x2 x3), = x1x2 x1x3

9.3.= x1(x2 x3), = x1x2 x1x3

9.4.= x1(x2 ! x3), = x1x2 ! x1x3

9.5.= x1(x2 j x3), = (x1x2) j (x1x3)

9.6.= x1(x2 # x3), = (x1x2) # (x1x3)

9.7.= x1 _ (x2x3), = (x1 _ x2)(x1 _ x3)

9.8.= x1 _ (x2 x3), = (x1 _ x2) (x1 _ x3)

9.9.= x1 _ (x2 x3), = (x1 _ x2) (x1 _ x3)

9.10.= x1 _ (x2 ! x3), = (x1 _ x2) ! (x1 _ x3)

9.11.= x1 _ (x2 j x3), = (x1 _ x2) j (x1 _ x3)

9.12.= x1 _ (x2 # x3), = (x1 _ x2) # (x1 _ x3)

9.13.= x1 (x2x3), = (x1 x2)(x1 x3)

9.14.= x1 (x2 _ x3), = (x1 x2) _ (x1 x3)

9.15.= x1 (x2 x3), = (x1 x2) (x1 x3)

9.16.= x1 (x2 ! x3), = (x1 x2) ! (x1 x3)

9.17.= x1 (x2 j x3), = (x1 x2) j (x1 x3)

9.18.= x1 (x2 # x3), = (x1 x2) # (x1 x3)

9.19.= x1 j (x2x3), = (x1 j x2) _ (x1 j x3)

9.20.= x1 j (x2 _ x3), = (x1 j x2)(x1 j x3)

9.21.= x1 j (x2 x3), = (x1 j x2) (x1 j x3)

9.22.= x1 j (x2 x3), = (x1 j x2) (x1 j x3)

9.23.= x1 j (x2 ! x3), = (x1 j x2) ! (x1 j x3)

9.24.= x1 j (x2 # x3), = (x1 j x2) # (x1 j x3)

9.25.= x1 # (x2x3), = (x1 # x2) _ (x1 # x3)

9.26.= x1 # (x2 _ x3), = (x1 # x2)(x1 # x3)

9.27.= x1 # (x2 x3), = (x1 # x2) (x1 # x3)

9.28.= x1 # (x2 x3), = (x1 # x2) (x1 # x3)

9.29.= x1 # (x2 ! x3), = (x1 # x2) ! (x1 # x3)

9.30.= x1 # (x2 j x3), = (x1 # x2) j (x1 # x3)

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10.Используя основные эквивалентности, доказать эквивалентность формул

è .

10.1.= (x3 j x2) j ((x3 # x1) _ x3), = (x1 # (x2 ! x1)) ! (x2 _ x3)

10.2.= x3 j ((x1 x2) x2), = (x2 # x3) j ((x1 x3) # x1)

10.3.= x2x3((x3 j (x3 # x2)) _ (x2 (x3 x1))), = (x2 j x3) # (x1x3)

10.4.= (x2 j x1) (x3 x1) (x1 _ x3), = x1 _ (x2 x3)

10.5.= (x3 (x2x3)) ((x2 _ x1)x1), = x1 (x3 ! x2)

10.6.= ((x1 x2) j (x2 _ x1)) (x3 j x2), = x1 (x2 ! x3)

10.7.= ((x3 x1) j x1) # (x3 j x2), = x1(x2x3 _ (x1 # x2))

10.8.= (x3 _ (x2 x1)) ! (x3 ! x1), = ((x2 x3) # (x2 _ x1))

10.9.= (x1x3) ! ((x1 x2) _ x2 _ x1), = (x1 # (x1 _ x3)) j (x1 _ x2x3)

10.10.= (x1(x3 x1)) # (x1 _ (x2 j x1)), = ((x1x1x2) ! (x3x1))

10.11.= (x2(x3 ! x2)) # ((x1 ! x2) _ x3), = x3(x1 j (x3 x2))

10.12.= (x2 ! x1) j (x3 _ x1 _ (x1 j x2)), = x1 (x1 _ (x1 x2))

10.13.= ((x2 x3) j (x1 x2))x3x1, = ((x2 # x1) _ (x1 x3))x1

10.14.= ((x1 ! x2) j x1)(x3 _ (x3 j x2)), = x1 ! ((x2 _ x3) ! x2)

10.15.= ((x2 x3)(x3 j x1)) j (x3 (x3 j x2)), = (x1 x3) j (x1 # x2)

10.16.= ((x3 _ x1) (x3 x2)) _ (x3 j x2), = x3 _ (x2 j x3) _ (x1 j x3)

10.17.= x3 j (x1 # (x2x3)), = x3 ! ((x1 _ x3) (x1x3 _ (x2 x3)))

10.18.= ((x1 x3) ! x1) _ ((x2 _ x3) x3), = (x2 # x3) j ((x3 x1) # x1)

10.19.= (x1 x3) j ((x3 ! (x1 j x2))x1), = (x1 ! (x2 # x3)) _ (x1 x3)

10.20.= x1 (x1 _ ((x2 x1) ! (x2 x1))), = (x2 ! x1)(x2(x1 _ x3) _ x2)

10.21.= (x1 ! x2) ((x3 _ x1) # (x3 j x1)), = ((x2x3) ((x1 x3)x2))x1

10.22.= (((x2x1) # (x1 j x2)) ! x1), = ((x1 x2) j ((x1 # x3)x3))

10.23.= (x1 j x3) j ((x1(x1 j x2)) _ x2), = x2 (x1x2) (x3 # x1)

10.24.= ((x3 ! x2) (x2 _ x3)) _ x3 _ x2, = ((x1 j x3) # (x1x3)) ! x2

10.25.= ((x1 # x2) (x3 # x1)) ! (x1 x2), = x3 _ (x2 # x3) _ (x1 x3)

10.26.= x3 ! ((x3 _ x1) ((x3x1) _ (x3 x2))), = x3 j ((x1 _ x3) j (x2 _ x1))

10.27.= ((x2 j x3) j x2)(x2 (x3 (x1x3))), = (x1x2x3 (x2 x3)) _ x1x2

10.28.= ((x2 j x1) _ (x3 (x2 # x1)))x1, = (((x1 j x3) # x2) x3)(x3 _ x1)

10.29.= (x3 _ x2 _ ((x1x3) (x3 ! x2))), = x3 # (((x2 ! x1) x3) _ x2)

10.30.= (((x2 x1) (x2 x3)) (x3 _ x2)), = (x1 x2) (x3 j x2)

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11.Эквивалентными преобразованиями привести формулу к ДНФ.

11.1.(x2 _ x1) j (x3 x1)

11.2.(x1 j x2) # (x3 j x1)

11.3.(((x2x1) _ x3) ! x3)

11.4.(((x1x2) ! x3) _ x2)

11.5.((x2 _ x3)x1) (x1 _ x2)

11.6.((x3 ! x1) # x2) j x1

11.7.x2 x3 ! (x1 _ x2)

11.8.((x2 j x3) # x3) # x1

11.9.x2 ! x3 j x1 x1

11.10.((x2 ! x1)x3) j x2

11.11.(x3 j x2) _ x1 # x3

11.12.((x1 _ x2) x3) # x1

11.13.(x1 _ (x3x2)) x2

11.14.(x3 j x2) (x1 _ x2)

11.15.(x1 ! x3) # (x2 ! (x1x3))

11.16.(x3 x2) ! (x1 j x3)

11.17.(x3 j x1) ! (x2 ! x1)(x2 ! x3)

11.18.x1 j ((x2 x3)x1x3)

11.19.(x1 _ x3) ! ((x2 _ x1) x2)

11.20.((x3 j x2) # (x1 # x2)) j x3

11.21.((x1 j x2)x2) _ ((x2 j x3)x2)

11.22.(x3 ! x2)x1 x3

11.23.(x2 j x3)(x1 j x3)x2

11.24.(x3 # x1) j (x2 x1)

11.25.(x1 j x2) # (x3 # x1)

11.26.(x1 # x3) ! ((x2 j x3)x1)

11.27.x3 (x1 ! (x2 # x3))

11.28.(x1 ! x2) (x1 x3)

11.29.(x1 _ x2) j ((x1x2) ! (x3 ! x2))

11.30.(x1 x2 x3) j x1

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12.Эквивалентными преобразованиями привести формулу к СДНФ.

12.1.x1 x2 (x1 _ x3)

12.2.(x1x2 # x3) x1

12.3.(x1 _ x2)(x3 j x1)

12.4.(x1 j x2) x1 x3

12.5.(x1 # x2) # (x1 x3)

12.6.x1 ((x2 x3) j x2)

12.7.x1 ((x2 # x3) # x3)

12.8.(x1 j x2) (x3 # x2)

12.9.(x1 _ x2)(x1 j x3)

12.10.x1 _ x2 _ x2 x3

12.11.((x1 j x2) _ x2) x3

12.12.x1 (x2 (x3 _ x2))

12.13.((x1 x2) ! x2) x3

12.14.(x1 _ (x2 ! x3)) x1

12.15.(x1 ! x2) ! (x1 # x3)

12.16.x1 ! x2 (x3 x1)

12.17.x1 ((x2 x3) # x3)

12.18.(x1 j x2)(x2 x3)

12.19.x1 (x2 # x1 x3)

12.20.(x1 _ x2) ! (x3 # x1)

12.21.x1 # ((x2 _ x3) x2)

12.22.(x1 j x2) x3 x2

12.23.(x1 x2)(x2 _ x3)

12.24.(x1 # x2)x2 # x3

12.25.(x1 ! x2) x1 x3

12.26.((x1 j x2) # x2) x3

12.27.((x1 # x2) # x3) # x1

12.28.x1 ((x2 x3) ! x3)

12.29.(x1 ! x2)(x3 x2)

12.30.x1(x2 x3)x2

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13.Эквивалентными преобразованиями привести формулу к КНФ.

13.1.x2((x3 ! x1) x2)

13.2.((x1 # x3) j x1)x2

13.3.((x3 x1) _ x2) x1

13.4.x3 j ((x2x1) j (x1x2x3))

13.5.((((x1 x3) _ x2)x2) x1)

13.6.((x1 # x3) _ (x2 j x2)x1)

13.7.(x1 _ x2)(x2 _ x3)(x2 _ x1)x1

13.8.x2 _ (x1 x3)(x1 _ x2)

13.9.((x1 x3)x2) x1

13.10.((x1 x3) ! x2) _ x2

13.11.(((x1 x2) x1) # x3)

13.12.x3 j ((x1x2) x1)

13.13.(((x3 _ x2)x1) x1)

13.14.(x3 j ((x1 ! x2))) j x3

13.15.((x1 # x2) j (x3x1))x3

13.16.(x2 ! x1)x2(x3 _ x2)

13.17.(x1 j x3)x2 x1

13.18.((x3x1) x2)(x1 x2)

13.19.(x1 _ (x2 x3) _ x3) x2

13.20.x1((x3 ! x2) x2)

13.21.((x1 x2)x3) _ x2

13.22.((x1 # x2) _ x3)x3

13.23.((x1 # (x1 j x3)) # x2)

13.24.(((x1 j x2) # x3) x3)

13.25.((x1(x3 # x2) _ x3) ! x2)

13.26.((x2 (x1 _ x3) x3) _ x1)

13.27.((x2 x3) j x1) # x3

13.28.x3((x1 ! x2) j (x2 # x1))

13.29.((x1 x2) j x3) # x2

13.30.((x1 _ x2)(x1 _ x3)) ! (x1 j x3)

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14.Эквивалентными преобразованиями привести формулу к СКНФ.

14.1.(x1 # x2) (x3 j x2)

14.2.x1 _ ((x2 x1) # x3)

14.3.(x1 j x2) (x1 _ x3)

14.4.(x1 ! x2)(x3 ! x2)

14.5.(x1x2 x3) x1

14.6.(x1 # (x1 x2)) # x3

14.7.(x1 _ x2) ! x2x3

14.8.(x1 (x2 x3)) x3

14.9.((x1 _ x2) x3) x1

14.10.x1x2 x2 x3

14.11.x1(x2 x2 j x3)

14.12.(x1 j x2) (x3 ! x1)

14.13.x1 ((x2 ! x1) # x3)

14.14.x1 (x2x3 # x2)

14.15.x1 ((x1 x2) _ x3)

14.16.(x1 j x2)x1x3

14.17.(x1 x2) (x2 # x3)

14.18.x1 ((x1 x2) # x3)

14.19.x1 ! x1 _ x2 ! x3

14.20.x1 x2 (x3 ! x1)

14.21.(x1 _ x2) j (x3 x1)

14.22.(x1 _ (x2 x3)) j x2

14.23.(x1 _ x2) x1x3

14.24.(x1 x2) (x3 j x1)

14.25.x1 ((x2 j x3) # x3)

14.26.(x1 _ x2) j x1 x3

14.27.(x1 ! x2)x1 x3

14.28.(x1 # x2) # x3x1

14.29.x1x2 x1 x3

14.30.x1 x2 (x1 j x3)

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15.Построить полином Жегалкина, используя эквивалентные преобразования.

15.1.x1 ! ((x2 _ x3) # x1)

15.2.(x1 # (x2 x3)) ! x2

15.3.x1 ((x2 x3) ! x3)

15.4.(x1x2 ! x3) x1

15.5.x1 # (x1 x2) ! x3

15.6.(x1 j (x2 _ x3)) x1

15.7.(x1 x2) _ (x1 # x3)

15.8.x1 # x2 # (x3 ! x1)

15.9.(x1 x2) # x3x1

15.10.x1 (x2 # (x1 j x3))

15.11.(x1 ! x2) (x1 j x3)

15.12.x1 ! (x1 ! x2 ! x3)

15.13.x1x2 # (x3 x1)

15.14.x1(x2 x1) # x3

15.15.(x1 x2 _ x3) _ x2

15.16.x1 ! x2 _ x2x3

15.17.(x1 j x2) # (x3 # x1)

15.18.(x1 ! (x1 x2)) # x3

15.19.(x1 _ (x2 # x1))x3

15.20.(x1 j x2) x3 x2

15.21.(x1 ! x2) x1 x3

15.22.x1 j (x2 (x2 ! x3))

15.23.(x1 x2) _ (x3 j x2)

15.24.x1 _ x2 _ (x3 # x2)

15.25.x1 j ((x2 # x3) j x1)

15.26.(x1 (x2 j x3)) # x1

15.27.(x1 # x2) ! x1x3

15.28.x1 # (x2 # x1 j x3)

15.29.(x1 j x2) ! (x3 _ x1)

15.30.(x1 x2) # (x3 x1)

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