=(?P?Q)?(Q??P)?T
(7)重言式
((P?Q)?(Q?P))?(P?Q)
=(P?Q)?(P?Q)?T
(8)重言式
P?(Q?R)?(P?Q?P?R)
=((P?Q)?(P?R))?(P?Q?P?R) =(P?Q?P?R)?(P?Q?P?R)
=T (9)重言式
P??P?Q =F?Q?T
(10)可满足式
P??Q?Q
=?(P??Q)?Q??P?Q?Q,当Q为真时公式为真,Q为假时公式为假。故为可
满足式。 (11)重言式
P?P?Q??P?P?Q?T
(12)重言式
P?Q?P??(P?Q)?P??P??Q?P?T
(13)可满足式
(P?Q?P)?(P?Q)的真值表如下: P F F T T (14)可满足式
Q F T F T P?Q?P T T F T P?Q T F F T (P?Q?P)?(P?Q) T F T T ((P?Q)?(R?S))?((P?R)?(Q?S))
=(?P?Q??R?S)?(?P??R?Q?S) =((?P??R)?(Q?S))?((?P??R)?(Q?S))
当Q或S有一个为真时公式为真;当Q和S均为假时,若P和R真值相同时,公式
为真;真值不同时,公式为假。故公式是可满足式。
2. 写出与下面给出的公式等价并且仅含有联接词?与?的最简公式。 (1)?(P?(Q?(R?P)))
??((P?(Q?(R?P)))?((Q?(R?P))?P))??((?P??Q?R?P)?((?Q?R?P)?P))??(T?(Q??R??P?P))??(Q??R??P?P)??P??(?P?Q??R)(2)((P?Q)?R)?(P?R)
?(?(P?Q)?R)?(P?R)??(?(P?Q)?R)?(P?R)?((P?Q)??R)?P?R?(P?Q?P)?(?R?P)?R
?((P?Q)?(?R?P))?R?(P?Q?R)?(?R?P?R)?(P?Q?R)??(?P??Q??R)(3)P?Q??R
??(?P??Q?R)
(4)P?(?Q?R?P)
?P?(?(?Q?R)?P)?P?(Q??R?P)
?P?Q??R??(?P??Q?R)(5)P?(Q?P)
?P?(?Q?P)??P?(?Q?P) ?T3. 写出与下面的公式等价并且仅含联结词?和?的最简公式。 (1)(P?Q)??P
?P?Q??P?F
(2)(P?(Q??Q))??P?Q
?(P?T)??P?Q?T??P?Q??P?Q??(P??Q)(3)?P??Q?(?R?P)
??P??Q?(R?P)?(?P??Q?R)?(?P??Q?P)?(?P??Q?R)?F??P??Q?R??(P?Q??R)4. 使用常用恒等式证明下列各式,并给出下列各式的对偶式。 (1)?(?P??Q)??(?P?Q)?P 证明:
?(?P??Q)??(?P?Q)?(P?Q)?(P??Q) ?P?(Q??Q)
?P?T?P对偶式:?(?P??Q)??(?P?Q)?P
(2)(P??Q)?(P?Q)?(?P??Q)??(?P?Q) 证明:
(P??Q)?(P?Q)?(?P??Q)?(P?(?Q?Q))?(?P??Q)?P?(?P??Q)
?(P??P)?(P??Q)?(P??Q)??(?P?Q)对偶式:(P??Q)?(P?Q)?(?P??Q)??(?P?Q)
(3)Q??((?P?Q)?P)?T 证明:
Q??((?P?Q)?P)?Q?(?(?P?Q)??P) ?Q?(P??Q)??P?(P??Q)??(P??Q)?T对偶式:Q??((?P?Q)?P)?F 5. 试证明下列合式公式是永真式。 (1)((P?Q)?P)?T 证明:
((P?Q)?P)??(P?Q)?P??P??Q?P?T(2)?(?(P?Q)??P)?F 证明:
?(?(P?Q)??P)??((P?Q)??P)??(P?Q)?P ??P??Q?P?F(3)(Q?P)?(?P?Q)?P 证明:
(Q?P)?(?P?Q)?(?Q?P)?(P?Q)?P?(?Q?Q) ?P?F?P(4)(P??P)?(?P?P)?F 证明:
(P??P)?(?P?P)?(?P??P)?(P?P)??P?P?F6. 证明下列蕴含式。 (1)P?Q?P?Q 证明:
(P?Q)?(P?Q)??(P?Q)?(?P?Q)??P??Q??P?Q ??P??Q?Q?T(2)P?(Q?R)?(P?Q)?(P?R) 证明:
(P?(Q?R))?((P?Q)?(P?R))?(?P?(?Q?R))?(?(?P?Q)?(?P?R))?(?P??Q?R)?((P??Q)??P?R)?(?P??Q?R)?(?P??Q?R)?T(3)P?Q?P?P?Q 证明:
(P?Q)?(P?P?Q)?(P?Q)?(?P?(P?Q))?(P?Q)?((?P?P)?(?P?Q))
?(P?Q)?(?P?Q)?(P?Q)?(P?Q)?T(4)(P?Q)?Q?P?Q 证明:
((P?Q)?Q)?(P?Q)?(?(?P?Q)?Q)?(P?Q)?((P??Q)?Q)?(P?Q)
?((P?Q)?(?Q?Q))?(P?Q)?(P?Q)?(P?Q)?T