求主合取范式,再用主合取范式求主析取.（p←→q）→r

Question

求主合取范式,再用主合取范式求主析取.
（p←→q）→r

Answer 1

(p←→q)→r
=[(p Λ q) v (¬p Λ ¬q)]→r
=[(p Λ q)→r] Λ [(¬p Λ ¬q)→r]
=[¬(p Λ q) V r] Λ [¬(¬p Λ ¬q) V r]
=[(¬p V ¬q V r] Λ [(p V q V r]
=M6 Λ M0 这个就是主合取范式
=m1 V m2 V m3 V m4 V m5 V m7 根据范式互补规律,这个就是主析取范式
主析取范式也可以硬性推算,比较麻烦
(p←→q)→r
=¬[(p Λ q) V (¬p Λ ¬q)] V r
=[(¬p V ¬q) Λ (p V q)] V r
=(¬p Λ p) V (¬p Λ q) V (¬q Λ p) V (¬q Λ q) V [r Λ (p V ¬p) Λ (q V ¬q)]
=(¬p Λ q) V (¬q Λ p) V [r Λ {(p Λ q) V (p V ¬q) V (¬p Λ q) V (¬p V ¬q)}]
=[¬p Λ q Λ (r V ¬r)] V [¬q Λ p Λ (r V ¬r)] V (p Λ q Λ r) V (p V ¬q Λ r) V (¬p Λ q Λ r) V (¬p V ¬q Λ r)
=(¬p Λ q Λ r) V (¬p Λ q Λ ¬r) V (p Λ ¬q Λ r) V (p Λ ¬q Λ ¬r) V (p Λ q Λ r) V (p V ¬q Λ r) V (¬p Λ q Λ r) V (¬p V ¬q Λ r)
=(¬p Λ q Λ r) V (¬p Λ q Λ ¬r) V (p Λ ¬q Λ r) V (p Λ ¬q Λ ¬r) V (p Λ q Λ r) V (¬p V ¬q Λ r)
=m3 V m2 V m5 V m4 V m7 V m1