matematika - semantiČka tablica i minimizacija
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Semantička tablica i minimizacijaTRANSCRIPT
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 �̅ ∧ �� ∧ � ̅
��� = ( � ∧ � ∧ �) ∨ (� ∧ �� ∧ �) ∨ (�̅ ∧ � ∧ �)̅ ∨ (�̅ ∧ �� ∧ �̅) (� ∧ �) ∨ (� ∧ ��) ∨ (�̅ ∧ �̅) ∨ ( � ∧ ��) (� ∧ �) ∨ (�̅ ∧ �)̅
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 = (�̅ ∨ �� ∨ �) ∧ (�̅ ∨ � ∨ �) ∧ (� ∨ � ∨ �)
(�̅ ∨ �) ∧ (�� ∨ �) ∧ (�̅ ∨ �) ∧ (� ∨ �)
(�̅ ∨ �) ∧ (� ∨ �)
� ∨ (�̅ ∧ �)
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 �̅ ∧ �� ∧ �̅
��� = ( � ∧ �� ∧ �) ∨ (�̅ ∧ �� ∧ �) ∨ (�̅ ∧ �� ∧ �̅)
= (� ∧ �̅) ∨ (�� ∧ �) ∨ (�̅ ∧ ��) ∨ (� ∧ �̅)
= (�� ∧ �) ∨ (�̅ ∧ ��)
= �� ∧ (� ∨ �̅)
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 � ∨ � ∨ �
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
��� = (�̅ ∨ �� ∨ �) ∧ (�̅ ∨ � ∨ �) ∧ (� ∨ �� ∨ �)
= (�̅ ∨ �) ∧ (�� ∨ �) ∧ (�̅ ∨ �) ∧ (�� ∨ �)
= (�̅ ∨ �) ∧ (�� ∨ �)
= � ∨ (�̅ ∧ ��)
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 � ∨ � ∨ �
��� = (�̅ ∨ �� ∨ �)̅ ∧ (�̅ ∨ �� ∨ �) ∧ (� ∨ � ∨ �)̅ ∧ (� ∨ � ∨ �)
= (�̅ ∨ ��) ∧ (�̅ ∨ �) ∧ (� ∨ �) ∧ (�̅ ∨ �)
= (�̅ ∨ ��) ∧ (� ∨ �)