matematika - semantiČka tablica i minimizacija

6
Zadaci iz matematike - semantička tablica i minimizacija A B 1.) (, , ) = ( ⟹ ) ∧ ⟺ (̅ ∨ ) x y z y ⟹ x (y ⟹ x) ∧ z ̅ ̅ ∧ B F baz.konjunkcija 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 ̅ ∧ ∧ 0 1 0 0 0 1 1 0 1 ̅ ∧ ∧ ̅ 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 ̅ ∧ = (̅ ∧ ∧ ) ∨ ( ∧ ∧ ̅) ∨ (̅ ∧ ∧ ) (̅ ∧) ∨ ( ∧ ̅)∨ (̅ ∧ ̅) ∨ ( ∧ ) (̅ ∧ ) ∨ (̅ ∧ ̅) ̅ ∧ ( ∨ ̅) A B 2.) (, , ) = ( ⟹ ) ∨ ( ∨ ) ∧ (̅ ⟺ ) x y z A ̅ ̅ ⟺ B A B baz.konjunkcije 1 1 1 0 0 1 1 0 0 1 1 ∧∧ 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 1 1 0 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 ̅ ∧∧̅ 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 ̅ ∧ ∧ ̅ = ( ∧ ∧ ) ∨ ( ∧ ∧ ) ∨ (̅ ∧ ∧ ̅) ∨ (̅ ∧ ∧ ̅) ( ∧ ) ∨ ( ∧ ) ∨ (̅ ∧ ̅) ∨ ( ∧ ) ( ∧ ) ∨ (̅ ∧ ̅)

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Semantička tablica i minimizacija

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Page 1: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

1.) � (�, �, �) = �(� ⟹ �) ∧ �� ⟺ (�̅ ∨ �)���������

x y z y ⟹ x (y ⟹ x) ∧ z �̅ �̅ ∧ � B F baz.konjunkcija

1 1 1 1 1 0 1 0 0

1 1 0 1 0 0 0 1 0

1 0 1 1 1 0 1 0 0

1 0 0 1 0 0 0 1 0

0 1 1 0 0 1 1 0 1 �̅ ∧ � ∧ �

0 1 0 0 0 1 1 0 1 �̅ ∧ � ∧ �̅

0 0 1 1 1 1 1 0 0

0 0 0 1 0 1 1 0 1 �̅ ∧ �� ∧ �

��� = (�̅ ∧ � ∧ �) ∨ (� � ∧ � ∧ �̅) ∨ (�̅ ∧ �� ∧ �)

(�̅ ∧ �) ∨ (� ∧ �̅) ∨ (�̅ ∧ �)̅ ∨ (� ∧ ��)

(�̅ ∧ �) ∨ (�̅ ∧ �̅)

�̅ ∧ (� ∨ �̅)

A B

2.) � (�, �, �) = �(� ⟹ ��) ∨ (�� ∨ �)� ∧ (�̅ ⟺ �)�����������

x y z �� � ⟹ �� �� ∨ � A �̅ �̅ ⟺ � B A ∧ B baz.konjunkcije

1 1 1 0 0 1 1 0 0 1 1 � ∧ � ∧ �

1 1 0 0 0 0 0 0 1 0 0

1 0 1 1 1 1 1 0 0 1 1 � ∧ �� ∧ �

1 0 0 1 1 1 1 0 1 0 0

0 1 1 0 1 1 1 1 1 0 0

0 1 0 0 1 0 1 1 0 1 1 �̅ ∧ � ∧ � ̅

0 0 1 1 1 1 1 1 1 0 0

0 0 0 1 1 1 1 1 0 1 1 �̅ ∧ �� ∧ � ̅

��� = ( � ∧ � ∧ �) ∨ (� ∧ �� ∧ �) ∨ (�̅ ∧ � ∧ �)̅ ∨ (�̅ ∧ �� ∧ �̅) (� ∧ �) ∨ (� ∧ ��) ∨ (�̅ ∧ �̅) ∨ ( � ∧ ��) (� ∧ �) ∨ (�̅ ∧ �)̅

Page 2: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

3.) � (�, �, �) = �(� ∧ �) ∨ (�̅ ∧ �̅)� ⟹ (� ⟺ �)

x y z � ∧ � �̅ �̅ �̅ ∧ � ̅ A B A⟹B baz. disjunkcije

1 1 1 1 0 0 0 1 1 1

1 1 0 1 0 1 0 1 0 0 �̅ ∨ � � ∨ �

1 0 1 0 0 0 0 0 0 1

1 0 0 0 0 1 0 0 1 1

0 1 1 0 1 0 0 0 1 1

0 1 0 0 1 1 1 1 0 0 � ∨ � � ∨ �

0 0 1 0 1 0 0 0 0 1

0 0 0 0 1 1 1 1 1 1

��� = ( �̅ ∨ � � ∨ �) ∧ (� ∨ � � ∨ �) = ( �̅ ∨ �) ∧ ( � � ∨ �) = ( � � ∨ �) A B

4.) � (�, �, �) = (�� ⟹ �̅��������� ⟹ �̅ ∨ �������) ⟺ ((�̅ ∨ �) ∧ �)

x y z �� �̅ �� ⟹ � ̅ �� ⟹ �̅��������� �̅ �̅ ∨ � �̅ ∨ ������� A B A⟺B baz. disj.

1 1 1 0 0 1 0 0 1 0 1 1 1

1 1 0 0 1 1 0 0 0 1 1 0 0 �̅ ∨ �� ∨ �

1 0 1 1 0 0 1 0 1 0 0 0 1

1 0 0 1 1 1 0 0 0 1 1 0 0 �̅ ∨ � ∨ �

0 1 1 0 0 1 0 1 1 0 1 1 1

0 1 0 0 1 1 0 1 1 0 1 1 1

0 0 1 1 0 0 1 1 1 0 0 0 1

0 0 0 1 1 1 0 1 1 0 1 0 0 � ∨ � ∨ �

KNF = (�̅ ∨ �� ∨ �) ∧ (�̅ ∨ � ∨ �) ∧ (� ∨ � ∨ �)

(�̅ ∨ �) ∧ (�� ∨ �) ∧ (�̅ ∨ �) ∧ (� ∨ �)

(�̅ ∨ �) ∧ (� ∨ �)

� ∨ (�̅ ∧ �)

Page 3: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

5.) �(�, �, �) = � � ∨ (� ∧ �)� ⟹ ((�̅ ∨ � ⟺ �)�����������������)

x y z � ∧ � � � ∨ (� ∧ �)� �̅ �̅ ∨ � �̅ ∨ � ⟺ � B F baz. disjunkcije

1 1 1 1 1 0 1 0 0

1 1 0 0 0 0 0 1 1 �̅ ∨ � � ∨ �̅

1 0 1 0 1 0 0 1 1

1 0 0 0 0 0 1 0 1

0 1 1 1 1 1 1 0 0

0 1 0 0 0 1 1 0 1 � ∨ � � ∨ �̅

0 0 1 0 1 1 0 1 1

0 0 0 0 0 1 0 1 1

KNF = (�̅ ∨ � � ∨ �̅) ∧ ( � ∨ � � ∨ �̅)

(�̅ ∨ �) ∧ (� � ∨ �̅)

(� � ∨ �̅)

A B

6.) �(�, �, �) = ( �̅ ∧ �� ) ⟹ � ) ⟺ (( � ⟺ �)̅ ∧ (�̅ ⟹ �̅))

x y z x� y� �̅ ∧ �� � �̅ � ⟺ �̅ �̅ ⟹ �̅ B F baz. konjunkcije

1 1 1 0 0 0 1 0 0 1 0 0

1 1 0 0 0 0 1 1 1 0 0 0

1 0 1 0 1 0 1 0 1 1 1 1 � ∧ �� ∧ �

1 0 0 0 1 0 1 1 0 0 0 0

0 1 1 1 0 0 1 0 0 1 0 0

0 1 0 1 0 0 1 1 1 0 0 0

0 0 1 1 1 0 1 0 1 1 1 1 �̅ ∧ �� ∧ �

0 0 0 1 1 1 0 1 0 0 1 1 �̅ ∧ �� ∧ �̅

��� = ( � ∧ �� ∧ �) ∨ (�̅ ∧ �� ∧ �) ∨ (�̅ ∧ �� ∧ �̅)

= (� ∧ �̅) ∨ (�� ∧ �) ∨ (�̅ ∧ ��) ∨ (� ∧ �̅)

= (�� ∧ �) ∨ (�̅ ∧ ��)

= �� ∧ (� ∨ �̅)

Page 4: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

7.) � = (�, �, �) = �̅ ∧ �������� ⟹ ((� ∨ �) ∧ (� ⟹ �� ) ∧ � ⟺ �̅��������� )

x y z x� �̅ ∧ � �̅ ∧ �������� � ∨ � �� � ⟹ �� (� ∨ �) ∧ (� ⟹ �� ) �̅ � ⟺ �̅ � ⟺ �̅��������� B F baz.konj

1 1 1 0 0 1 1 0 0 0 0 0 1 0 0

1 1 0 0 0 1 1 0 0 0 1 0 1 0 0

1 0 1 0 0 1 1 1 1 1 0 0 1 1 1 � ∧ �� ∧ �

1 0 0 0 0 1 1 1 1 1 1 0 1 1 1 � ∧ �� ∧ �̅

0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 �̅ ∧ � ∧ �

0 1 0 1 1 0 1 0 1 1 1 0 1 1 1 �̅ ∧ � ∧ �̅

0 0 1 1 0 1 0 1 1 0 0 0 1 0 0

0 0 0 1 0 1 0 1 1 0 1 0 1 0 0

��� = (� ∧ �� ∧ �) ∨ (� ∧ �� ∧ �̅) ∨ (�̅ ∧ � ∧ �) ∨ (�̅ ∧ � ∧ �̅)

= (� ∧ �� ) ∨ (� ∧ �̅) ∨ (�̅ ∧ �) ∨ (� ∧ �)̅

= (� ∧ �� ) ∨ (�̅ ∧ �)

A B C

8.) �(�, �, �) = (� ⟹ �����������) ∧ ( � ∧ �̅) ∨ ( �̅ ⟺ � )

��� = ( � ∧ � ∧ �)̅ ∨ (� ∧ �� ∧ �̅) ∨ (�̅ ∧ � ∧ �) ∨ (�̅ ∧ �� ∧ �)

= (� ∧ �̅) ∨ (� ∧ ��) ∨ (�̅ ∧ �) ∨ (� ∧ ��)

= (� ∧ �̅) ∨ (�̅ ∧ �)

��� = (�̅ ∨ �� ∨ �̅) ∧ (�̅ ∨ � ∨ �̅) ∧ (� ∨ �� ∨ �) ∧ (� ∨ � ∨ �)

= (�̅ ∨ �̅) ∧ (�� ∨ �) ∧ (� ∨ �) ∧ (�� ∨ �)

= (�̅ ∨ �̅) ∧ (� ∨ �)

x y z �� � ⟹ �� (� ⟹ �����������) �̅ � ∧ �̅ �̅ �̅ ⟺ � � ∧ � (� ∧ �) ∨ � baz. konj. baz.disj

1 1 1 0 0 1 0 0 0 0 0 0 �̅ ∨ �� ∨ �̅

1 1 0 0 0 1 1 1 0 1 1 1 � ∧ � ∧ �̅

1 0 1 1 1 0 0 0 0 0 0 0 �̅ ∨ � ∨ �̅

1 0 0 1 1 0 1 0 0 1 0 1 � ∧ �� ∧ �̅

0 1 1 0 1 0 0 0 1 1 0 1 �̅ ∧ � ∧ �

0 1 0 0 1 0 1 1 1 0 0 0 � ∨ �� ∨ �

0 0 1 1 1 0 0 0 1 1 0 1 �̅ ∧ �� ∧ �

0 0 0 1 1 0 1 0 1 0 0 0 � ∨ � ∨ �

Page 5: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

9.) �(�, �, �) = � �̅ ∧ (�̅ ⟹ �)� ⟺ ( � ∧ �������)

x y z x� �̅ �̅ ⟹ � A � ∧ � � ∧ ������� F baz. disjunkcije

1 1 1 0 0 1 0 1 0 1

1 1 0 0 1 1 0 0 1 0 �̅ ∨ �� ∨ �

1 0 1 0 0 1 0 1 0 1

1 0 0 0 1 0 0 0 1 0 �̅ ∨ � ∨ �

0 1 1 1 0 1 1 0 1 1

0 1 0 1 1 1 1 0 1 1

0 0 1 1 0 1 1 0 1 1

0 0 0 1 1 0 0 0 1 0 � ∨ � ∨ �

��� = (�̅ ∨ �� ∨ �) ∧ (�̅ ∨ � ∨ �) ∧ (� ∨ � ∨ �)

= (�̅ ∨ �) ∧ (� ∨ ��) ∧ (�̅ ∨ �) ∧ (� ∨ �)

= (�̅ ∨ �) ∧ (� ∨ �)

= � ∨ (�̅ ∧ �)

A B

10.) �(�, �, �) = �(� ⟹ �) ∧ ( (�� ∧ �̅ ) ∨ ��

A B

x y z � ⟹ � �� �̅ �� ∧ �̅ (�� ∧ �̅) ∨ � � ∧ � baz. disjunkcije

1 1 1 1 0 0 0 1 1

1 1 0 0 0 0 0 0 0 �̅ ∨ �� ∨ �

1 0 1 1 1 0 0 1 1

1 0 0 0 1 0 0 0 0 �̅ ∨ � ∨ �

0 1 1 1 0 1 0 1 1

0 1 0 1 0 1 0 0 0 � ∨ �� ∨ �

0 0 1 1 1 1 1 1 1

0 0 0 1 1 1 1 1 1

��� = (�̅ ∨ �� ∨ �) ∧ (�̅ ∨ � ∨ �) ∧ (� ∨ �� ∨ �)

= (�̅ ∨ �) ∧ (�� ∨ �) ∧ (�̅ ∨ �) ∧ (�� ∨ �)

= (�̅ ∨ �) ∧ (�� ∨ �)

= � ∨ (�̅ ∧ ��)

Page 6: MATEMATIKA - SEMANTIČKA TABLICA I MINIMIZACIJA

Zadaci iz matematike - semantička tablica i minimizacija

A B

11.) �(�, �, �) = �(�̅ ⟹ �) ∧ (� ⟹ �̅)� ∧ ((� ⟺ �̅ ) ⟹ (� ∨ �)̅

x y z �̅ �̅ ⟹ � � � ⟺ �̅ �̅ � ∨ �̅ � F baz. disjunkcije

1 1 1 0 1 0 0 0 1 1 0 �̅ ∨ �� ∨ �̅

1 1 0 0 1 0 0 1 1 1 0 �̅ ∨ �� ∨ �

1 0 1 0 1 1 0 0 0 1 1

1 0 0 0 1 1 0 1 1 1 1

0 1 1 1 1 1 0 0 1 1 1

0 1 0 1 1 1 0 1 1 1 1

0 0 1 1 0 0 0 0 0 1 0 � ∨ � ∨ �̅

0 0 0 1 0 0 0 1 1 1 0 � ∨ � ∨ �

��� = (�̅ ∨ �� ∨ �)̅ ∧ (�̅ ∨ �� ∨ �) ∧ (� ∨ � ∨ �)̅ ∧ (� ∨ � ∨ �)

= (�̅ ∨ ��) ∧ (�̅ ∨ �) ∧ (� ∨ �) ∧ (�̅ ∨ �)

= (�̅ ∨ ��) ∧ (� ∨ �)