离散数学：第一次作业

题目 P97/3 #

确定下列个命题的真假性:

[leftmargin=1.5cm]

• (1) $\varnothing \subseteq \varnothing$;
• (2) $\varnothing \in \varnothing$;
• (3) $\varnothing \subseteq \{\varnothing\}$;
• (4) $\varnothing \in \{\varnothing\}$;
• (5) $\{a,b\} \subseteq \{a,b,c,\{a,b,c\}\}$;
• (6) $\{a,b\} \in \{a,b,c,\{a,b,c\}\}$;
• (7) $\{a,b\} \subseteq \{a,b,\{\{a,b,c\}\}\}$;
• (8) $\{a,b\} \in \{a,b,\{\{a,b,c\}\}\}$;

答案

(1) 真 (2) 假 (3) 真 (4) 真 (5) 真 (6) 假 (7) 真 (8) 假

题目 P97/4 #

对任意集合 $A,B,C$，确定下列命题的真假性:

[leftmargin=1.5cm]

• (1) 如果 $A\not\in B \wedge B \not\in C$，则 $A\not\in C$;
• (2) 如果 $A\in B \wedge B \not\in C$，则 $A\not\in C$;
• (3) 如果 $A\subseteq B \wedge B \not\in C$，则 $A\not\in C$.

答案

(1) 假 (2) 假 (3) 假

题目 P97/5 #

对任意集合 $A,B,C$，确定下列命题的真假性:

[leftmargin=1.5cm]

• (1) 如果 $A\in B \wedge B \subseteq C$，则 $A\in C$;
• (2) 如果 $A\in B \wedge B \subseteq C$，则 $A\subseteq C$;
• (3) 如果 $A\subseteq B \wedge B \in C$，则 $A\in C$.

答案

(1) 真 (2) 假 (3) 假

题目 P98/6 #

求下列集合的幂集:

[leftmargin=1.5cm]

• (1) $\{a,b,c\}$;
• (2) $\{a,\{b,c\}\}$;
• (3) $\{\varnothing\}$;
• (4) $\{\varnothing,\{\varnothing\}\}$.

答案

(1) $\{\varnothing,\{a\},\{b\},\{c\},\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$;

(2) $\{\varnothing,\{a\},\{\{b,c\}\},\{a,\{b,c\}\}\}$;

(3) $\{\varnothing,\{\varnothing\}\}$;

(4) $\{\varnothing,\{\varnothing\},\{\{\varnothing\}\},\{\varnothing,\{\varnothing\}\}\}$.

题目 P98/8 #

设 $A,B,C$ 是集合, 证明:

[leftmargin=1.5cm]

• (1) $(A \backslash B)\backslash C = A\backslash(B \cup C)$;
• (2) $(A \backslash B)\backslash C = (A\backslash C)\backslash(B \backslash C)$;
• (3) $(A \backslash B)\backslash C = (A \backslash C)\backslash B$;

证明

\

我们记 $D=A\cup B\cup C$ 为全集.

(1) $(A\backslash B) \backslash C = (A\cap B')\cap C' = A \cap (B' \cap C') = A\cap(B \cup C)' = A\backslash (B \cup C)$;

(2) $(A\backslash B) \backslash C = (A\cap B')\cap C' = (A \cap C') \cap (B' \cup C) = (A \backslash C)\cap(B \cap C')' \\= (A \backslash C)\cap (B \backslash C)' = (A \backslash C)\backslash (B \backslash C)$;

(3) $(A\backslash B) \backslash C = (A\cap B')\cap C' = (A \cap C') \cap B' = (A \backslash C)\cap B'= (A \backslash C)\backslash B$.

题目 P98/9 #

设 $A,B$ 是集合 $X$ 的子集, 证明:

$$A \subseteq B \Leftrightarrow A' \cup B = X \Leftrightarrow A\cap B' = \varnothing$$

先证明 $A \subseteq B \Leftrightarrow A' \cup B = X$

证明

$"\Rightarrow"$: $A'\cup B = A' \cup (A \cup B) = A'\cup A \cup B = X \cup B = X$

$"\Leftarrow"$ $A' \cup B = X \Rightarrow (X \backslash A') \subseteq B \Rightarrow A \subseteq B$

再证明 $A \subseteq B \Leftrightarrow A\cap B' = \varnothing$

证明

$"\Rightarrow"$: $A\cap B' = A \cap (X \backslash B) = (A \cap X) \backslash B = A \backslash B = \varnothing$

$"\Leftarrow"$ $A \cap B' = \varnothing \Rightarrow A \subseteq (X \backslash B') \Rightarrow A \subseteq B$

题目 P98/10 #

对于任意集合 $A,B,C$, 下列各式是否成立, 为什么?

• (1) $A \cup B = A\cup C \Rightarrow B = C$;
• (2) $A \cap B = A \cap C \Rightarrow B = C$.

答案

• (1) 不成立, 例如取 $A=\{1,2\},B=\{1\},C=\{2\}$.
• (2) 不成立, 例如取 $A=\{1\},B=\{1,2\},C=\{1,3\}$