结构推理
用真值表方法证明下面各题：
(1)P→(Q→R)=Q→(P→R)．
(2)(P→Q)∧(R→Q)=(P∨R→Q)．
(3)P→Q=￢P∨Q．
【正确答案】(1)将等式两边分别构造真值表：
| P | Q | R | Q→R | P→(Q→R) |
| F | F | F | T | T |
| F | F | T | T | T |
| F | T | F | F | T |
| F | T | T | T | T |
| T | F | F | T | T |
| T | F | T | T | rr |
| T | T | F | F | F |
| T | T | T | T | T |
| P | Q | R | P→R | Q→(P→R) |
| F | F | F | T | T |
| F | F | T | T | T |
| F | T | F | T | T |
| F | T | T | T | T |
| T | F | F | F | T |
| T | F | T | T | T |
| T | T | F | F | F |
| T | T | T | T | T |
比较两表的最右列知其真假取值一致．
故有：P→(Q→R)=Q→(P→R)．
(2)分别构造等号两边的真值表：
| P | Q | R | P→Q | R→Q | (P→Q)∧(R→Q) |
| F | F | F | T | T | T |
| F | F | T | T | F | F |
| F | T | F | T | T | T |
| F | T | T | T | T | T |
| T | F | F | F | T | F |
| T | F | T | F | F | F |
| T | T | F | T | T | T |
| T | T | T | T | T | T |
| P | Q | R | P∨R | P∨R→Q |
| F | F | F | F | T |
| F | F | T | T | F |
| F | T | F | F | T |
| F | T | T | T | T |
| T | F | F | T | F |
| T | F | T | T | F |
| T | T | F | T | T |
| T | T | T | T | T |
比较两表最右列知其真值表一致．
故有：(P→Q)∧(R→Q)=(P∨R→Q)．
(3)分别构造等式两边的真值表：
| P | Q | P→Q |
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
| P | Q | ￢P | ￢P∨Q |
| F | F | T | T |
| F | T | T | T |
| T | F | F | F |
| T | T | F | T |
比较两表最右列知其真值表一致．
故有：P→Q=￢P∨Q．
【答案解析】