利用常用的逻辑等价式证明:

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利用常用的逻辑等价式证明:(P ∨ ((P∧(P ∨~Q)) ∧ ~Q)) ∧ (Q →(Q∧~P)) ≡ (P ∧ ~Q)

热搜:先导式溢流阀的主阀芯是利用什么作用开启的

答案解析

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  • [证明]由分配律可知:
       (P∧Q)∨(P∧¬Q)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DPlatex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(Q∨¬Q)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DP∧1latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DP$由吸收律可知:P∨(P∧Q)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DP,所以
       Q→(P∨(P∧Q))latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DQ→P$P→(P→Q)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬P∨(P→Q)
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3DP∨¬P∨Q
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬P∨Q
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬¬Q∨¬P
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬Q→¬P$用真值表可以证明f6.184CB7E.jpg,于是有
       f6.18510C4.jpg$A→(B∨C)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬A∨B∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬(A∧¬B)∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(A∧¬B)→C$(A→C)∧(B→C)latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬A∨C)∧(¬B∨C)
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬A∧¬B)∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬(A∨B)∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(A∨B)→C$((A∧B)→C)∧(B→(D∨C))
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬(A∧B)∨C)∧(¬B∨D∨C)
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬A∨¬B∨C)∧(¬B∨D∨C)
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D((¬A∨¬B)∧(¬B∨D))∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D((¬A∧D)∨¬B)∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬(A∨¬D)∨¬B)∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬(B∧(¬D∨A))∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬(B∧(D→A))∨C
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(B∧(D→A))→C$由(4)的证明结果可知:
       右式latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(C∧((A∧B)∨(¬A∧¬B)))→D
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D((C∧A∧B)∨(C∧¬A∧¬B))→D
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D¬((C∧A∧B)∨(C∧¬A∧¬B))∨D
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬(C∧A∧B)∧¬(C∧¬A∧¬B))∨D
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬(C∧A∧B)∧(¬C∨A∨B))∨D
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D(¬(C∧A∧B)∨D)∧(¬C∨A∨B∨D)
       latex.action?latex=XExlZnRyaWdodGFycm93IA%3D%3D((C∧A∧B)→D)∧(C→(A∨B∨D))