Section 6.10 消去量词的练习题
Objectives: 给定解释\(\quad I \quad\)如下
- 设个体域\(D =\{2,3\} \)
- \(\displaystyle \bar{L}(x,y) \text{为}\)
- \(\displaystyle \quad \bar{L}(2,2) = \bar{L}(3,3) =1 \)
- \(\displaystyle \quad \bar{L}(2,3)= \bar{L}(3,2) = 0 \)
Exercises Exercises
1.
在解释I下求\(\forall x \exists y L(x,y) \)的值
Hint
Solution
按从内到外的顺序消去量词
\begin{align*}
\amp \forall x \exists y L(x,y) \\
\amp \Iff \forall x (\bar{L}(x,2) \vee \bar{L}(x,3)) \\
\amp \Iff [(\bar{L}({\color{Blue}{2}},2) \vee \bar{L}({\color{Blue}{2}},3))] \wedge [(\bar{L}({\color{green}{3}},2) \vee \bar{L}({\color{green}{3}},3))]\\
\amp \Iff (1 \vee 0) \wedge (0 \vee 1) \\
\amp \Iff 1
\end{align*}
Exercises Exercises
2.
在解释I下求\(\exists y \forall x L(x,y) \)的值
Hint
Solution
按从内到外的顺序消去量词
\begin{align*}
\amp \exists y \forall x L(x,y) \\
\amp \Iff \exists y [\bar{L}(2,y) \wedge \bar{L}(3,y)] \\
\amp \Iff [\bar{L}(2,2) \wedge \bar{L}(3,2)] \vee [\bar{L}(2,3) \wedge \bar{L}(3,3)]\\
\amp \Iff (1 \wedge 0) \vee (0 \wedge 1) \\
\amp \Iff 0 \vee 0 \\
\amp \Iff 0
\end{align*}