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第一次作业

题目 P97/3

确定下列个命题的真假性:

(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) \(\{a,b\} \in \{a,b,\{\{a,b,c\}\}\}\);

解:

• (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:

(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 \(A\not\in C\).

解:

• (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:

(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 \(A\in C\).

解:

• (1) 真
  (2) 假
  (3) 假

题目 P98/6

求下列集合的幂集:

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

解:

• (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:

(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) \(A \cap B = A \cap C \Rightarrow B = C\).

解:

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