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来源：学生作业帮助网 编辑：作业帮 时间：2024/05/10 06:42:06
是全部
是全部
是全部
1
证 令P(x)：x是有理数,Q(x)：x是实数,R(x)：x是整数,则
(∀x)(P(x)→Q(x))⋀(∃x)(P(x)⋀R(x))⇒(∃x)(Q(x)⋀R(x))
(1) (∃x)(P(x)⋀R(x) ) P
(2) R(y)⋀Q(y) ES,(1),y被固定
(3) (∀x)(P(x)→Q(x) ) P
(4) P(y)→Q(y) US,(3)
(5) P(y) T,(1)
(6) Q(y) T,(4)(5)
(7) R(y) T,(1)
(8) Q(y)⋀R(y) T,(6)(7)
(9) (∃x)(Q(x)⋀R(x)) EG,(7)
2
证 令P(x)：x是大学生,Q(x)：x是文科生,R(x)：x是理科生,S(x):x是优等生,a：小张,则
(∀x)(P(x)→Q(x)⋁R(x))⋀(∃x)(P(x)⋀S(x))⋀(¬R(a))⋀S(a)⇒P(a)→Q(a)
(1) (∃x)(P(x)⋀S(x) ) P
(2) P(a)⋀S(a) ES,(1),a被固定
(3) S(a) P
(4) P(a) T,(3)
(5) (∀x)(P(x)→Q(x)⋁R(x)) P
(6) P(a)→Q(a)⋁R(a) US,(5)
(7) Q(a)⋁R(a) T,(4)(6)
(8) ¬R(a) P
(9) Q(a) T,(7)(8)
(10) ¬P(a)⋁Q(a) T,(4)(9)
(11) P(a)→Q(a) E16（条件式转化）
3
证 令P(x)：x是偶数,Q(x)：x能被2整除,a：8,则
(∀x)(P(x)→Q(x))⋀P(8)⇒Q(8)
(1) P(8) P
(2) (∀x)(P(x)→Q(x)) P
(3) P(8)→Q(8) US,(2)
(4) Q(8) T,(1)(3)
4
证 令P(x)：x是大学生,Q(x)：x是勤奋的,a：小李,则
(∀x)(P(x)→Q(x))⋀(¬Q(a))⇒¬P(a)
(1) ¬Q(a) P
(2) (∀x)(P(x)→Q(x) ) P
(3) P(a)→Q(a) US,(2)
(4) ¬P(a) I12(拒取式),(1)(4)