离散数学 第二章 谓词逻辑 习题课

6)判断下面推证是否正确。
(x)(A(x)→B(x)) ⑴ (x)(A(x)∨B(x)) ⑵ (x)(A(x)∧B(x) ⑶ (x)(A(x)∧B(x)) ⑷ ((x)A(x)∧(x)B(x)) ⑸ (x)A(x)∨(x)B(x) ⑹ (x)A(x)∨(x)B(x) ⑺ (x)A(x)→(x)B(x) 第⑷步错，由⑶到⑷用的是公式： (x)(A(x)∧B(x))((x)A(x)∧(x)B(x)) 无 此 公 式 ， 而 是 (x)(A(x)∧B(x)) (x)A(x)∧(x)B(x)，应将⑷中的换成 即：
习题课
62页
(2)
(x)y((P(x)∧P(y)∧E(x,y)) →z(L(z)∧R(x,y,z)∧t((L(t)∧R(x,y,t))→E(t,z))))
(3)b)设R(x):x是实数，G(x,y):x＞y
(x)(R(x)→y(R(y)∧G(y,x))) c)设R(x):x是实数，G(x,y):x＞y f(x,y)=x+y g(x,y)=xy (x)yz(R(x)∧R(y)∧R(z)∧G(f(x,y),g(x,z))) 或者 (x)yz(R(x)∧R(y)∧R(z)∧G(x+y,xz))
离 散 数 学
第二章 谓词逻辑 习题课
一. 命题符号化 60页(2)
a) (x)(J(x)→L(x)) b) (x)(L(x)∧S(x)) c) (x)(J(x)∧O(x)∧V(x)) d) J(j)∧O(j)∧V(j) e) (x)(L(x)→J(x)) 或者 (x)(L(x)∧J(x) f) (x)(S(x)∧L(x)∧C(x)) g) (x)(C(x)∧V(x) 或者(x)(C(x)→V(x)) h) (x)((C(x)∧O(x))→L(x)) i) (x)(W(x)∧C(x)∧H(x)) j) (x)(W(x)∧J(x)∧C(x)) k) (x)(L(x)→y(J(y)∧A(x,y))) l) (x)(S(x)∧y(L(y)→A(x,y)))
习题课
5)b)设N(x):x是数，A(x,y):y是x的后继数
(x)(N(x)∧A(x,1))
(6) 设 A(x):x 是戴眼镜的 ,B(x):x 是用功的 ,C(x):x 是大 学生,D(x):x是大的,E(x):x是厚的，F(x):x是巨著, A(x,y):x在看y,a:那位，b:这本 A(a)∧B(a)∧C(a)∧D(b)∧E(b)∧F(b)∧ A(a,b)
75页
(1)b)(x)(yP(x,y)→(zQ(z)→R(x))) (x)(yP(x,y)∨(zQ(z)∨R(x))) (x)(yP(x,y)∨(zQ(z)∨R(x))) (x)(yP(x,y)∨ z(Q(z)∨R(x))) (x)yz(P(x,y)∨(Q(z)∨R(x))) (2)c)(x)P(x)→(x)(zQ(x,z)∨zR(x,y,z)) (x)P(x)∨(x)(zQ(x,z)∨zR(x,y,z)) (x)P(x)∨(x)(zQ(x,z)∨zR(x,y,z)) (x)P(x)∨u(zQ(u,z)∨tR(u,y,t)) (x)uzt(P(x)∨(Q(u,z)∨R(u,y,t))) (x)uzt(P(x)∨Q(u,z)∨R(u,y,t)) 此式既是前束析取范式，也是前束合取范式。
66页
(3)b)P:2>1,Q(x):x≤3, R(x):x>5,a:5,{-2,3,6} (x)(P→Q(x))∨R(a)(P→(x)Q(x))∨R(a) (P→(Q(-2)∧Q(3)∧Q(6)))∨R(5) (T→(T ∧T ∧F ))∨F (T→F)∨FF∨F F 4)b)对约束变元换名 (x)(P(x)→(R(x)∨Q(x)))∧ (x)R(x)→zS(x,z) y(P(y)→(R(y)∨Q(y)))∧ tR(t)→uS(x,u) (5)a)对自由变元代入 (yA(x,y)→(x)B(x,z))∧ (x)zC(x,y,z) (yA(u,y)→(x)B(x,v))∧ (x)zC(x,w,z)
习题课
72页(2)d)论域为{1,2} P(1) P(2) Q(1,1) Q(1,2) Q(2,1) Q(2,2) F T T T F F
(x)y(P(x)∧Q(x,y)) y(P(1)∧Q(1,y))∧y(P(2)∧Q(2,y)) ((P(1)∧Q(1,1))∨(P(1)∧Q(1,2)))∧ ((P(2)∧Q(2,1))∨(P(2)∧Q(2,2))) ((F∧T)∨(F∧T))∧((T∧F)∨(T∧F)) (F∨F)∧(F∨F)F
Hale Waihona Puke Baidu充题：
1.每个人的叔叔都是他父亲的弟弟。 设：P(x):x是人，U(x,y):y是x的叔叔， B(x,y):x是y的弟弟， f(x)=x的父亲 (x)(P(x)→y(U(x,y)→B(y,f(x))) 2.下面是判定一个年号是否为闰年的命题: “年号能被4整除并且不能被100整除的为闰年. 或者年 号能被400整除的也是闰年.” 设 Y(x):x是年号; D(x,y):x可整除y; R(x):x是闰年 (x)(Y(x)→(((D(4,x)∧D(100,x))→R(x))∨(D(400,x) →R(x))))
(x)(A(x)→B(x))
(x)(A(x)→B(x)) ⑴ (x)(A(x)∨B(x)) ⑵ (x)(A(x)∧B(x) ⑶ (x)(A(x)∧B(x)) ⑷ ((x)A(x)∧(x)B(x)) ⑸ (x)A(x)∨(x)B(x) ⑹ (x)A(x)∨(x)B(x) ⑺ (x)A(x)→(x)B(x) 因为由公式E18 P→QQ→P (x)(A(x)∧B(x)) (x)A(x)∧(x)B(x) ， P Q 得 ((x)A(x)∧(x)B(x))(x)(A(x)∧B(x))