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Discrete mathematics: Main Normal Form (main disjunctive normal form, main conjunctive normal form)
2022-07-05 00:58:00 【Xiaobai Xiaozheng】
Solve the main disjunctive normal form 、 Principal conjunctive normal form method
1、 Truth table method
① List all possible values of arguments in the table
② Look up the table to judge the proposition
Propositional result really , The argument value corresponds to Principal disjunctive normal form
Propositional result false , The argument value corresponds to The main conjunctive paradigm
2、 Equivalent algorithm
① Propositional simplification
Implication equivalence :A→B ¬A∨B( effect : Go to →)
Law of contradiction :A ∧¬A( effect : Complement argument )
Distributive law :(A∧B)∨C (A∨C)∧(B∨C)、(A∨B)∧C (A∧C)∨(B∧C)
② Judgement proposition
Propositional result really , The argument value corresponds to Principal disjunctive normal form
Propositional result false , The argument value corresponds to The main conjunctive paradigm
Example
Find the formula (p→q)∧(q→r) Principal disjunctive normal form and principal conjunctive normal form of 、 True assignment .
Explain :
1、 Truth table method
p | q | r | p→q | q→r | (p→q)∧(q→r) |
---|---|---|---|---|---|
0 | 0 | 0 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 0 | 0 |
0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 1 | 0 |
1 | 1 | 0 | 1 | 0 | 0 |
1 | 1 | 1 | 1 | 1 | 1 |
Look up the table
True assignment :000、001、011、111
Principal disjunctive normal form :∑(0,1,3,7)
False assignment :010、100、101、110
The main conjunctive paradigm :∏(2,4,5,6)
2、 Equivalent algorithm
(p→q)∧(q→r)
= (¬p∨q)∧(¬q∨r) -----------------------------( Implication equivalence : Simplification →)
= ((¬p∨q)∨(¬r∧r)))∧((¬q∨r)∨(¬p∧p))----( Law of contradiction : Complement argument )
= (¬p∨q∨¬r)∧(¬p∨q∨r)∧(¬p∨¬q∨r)∧(p∨¬q∨r)—( Distributive law : Simplification )
M5M4M6M2
According to the calculation results The main conjunctive paradigm :∏(2,4,5,6)
False assignment :010、100、101、110
be Principal disjunctive normal form :∑(0,1,3,7)
True assignment :000、001、011、111
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