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zt5JV)Um@$;?(1Y@!F?_I6JLh6eOL$n0}gpXD+*<9WOHB}Gq03RFd?O8@`> diff --git a/2021/solution/hw7-function-ordering-solution/hw7-function-ordering-solution.tex b/2021/solution/hw7-function-ordering-solution/hw7-function-ordering-solution.tex index 374c8c0..ab543cc 100644 --- a/2021/solution/hw7-function-ordering-solution/hw7-function-ordering-solution.tex +++ b/2021/solution/hw7-function-ordering-solution/hw7-function-ordering-solution.tex @@ -10,7 +10,7 @@ %%%%%%%%%%%%%%%%%%%% \title{7. 集合: 函数与序关系 (7-function-ordering)} \me{魏恒峰}{hfwei@nju.edu.cn}{}{} -\date{2021年04月22日 发布作业 \\ 2021年05月xx日 发布答案} +\date{2021年04月22日 发布作业 \\ 2021年06月05日 发布答案} %%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle @@ -40,6 +40,34 @@ \end{problem} \begin{proof} + \begin{enumerate}[(1)] + \item 对于任意 $b \in B$, + \begin{align} + &b \in f(A_{1} \cup A_{2}) \\ + \iff & \exists a \in A_{1} \cup A_{2}.\; f(a) = b \\ + \iff & (\exists a \in A_{1}.\; f(a) = b) \lor (\exists a \in A_{2}.\; f(a) = b) \\ + \iff & b \in f(A_{1}) \lor b \in f(A_{2}) \\ + \iff & b \in f(A_{1}) \cup f({A_{2}}) + \end{align} + \item 对于任意 $a \in A$, + \setcounter{equation}{0} + \begin{align} + &a \in f^{-1}(B_{1} \setminus B_{2}) \\ + \iff & f(a) \in B_{1} \setminus B_{2} \\ + \iff & f(a) \in B_{1} \land f(a) \notin B_{2} \\ + \iff & a \in f^{-1}(B_{1}) \land \lnot(a \in f^{-1}(B_{2})) \\ + \iff & a \in f^{-1}(B_{1}) \land a \notin f^{-1}(B_{2}) \\ + \iff & a \in f^{-1}(B_{1}) \setminus f^{-1}(B_{2}) + \end{align} + \item 对于任意 $b \in B$, + \setcounter{equation}{0} + \begin{align} + &b \in f(f^{-1}(B_{0})) \\ + \iff & \exists a \in f^{-1}(B_{0}).\; b = f(a) \\ + \iff & \exists a \in A.\; f(a) \in B_{0} \land b = f(a) \\ + \implies & b \in B_{0} + \end{align} + \end{enumerate} \end{proof} %%%%%%%%%%%%%%% @@ -54,6 +82,36 @@ \end{problem} \begin{proof} + \begin{enumerate}[(1)] + \item 任取 $c \in C$。 + 由于 $g$ 是满射, + \[ + \exists b \in B.\; g(b) = c. + \] + 任取 $b_{0}$ 满足 $g(b_{0}) = c$。 + 由于 $f$ 是满射, + \[ + \exists a \in A.\; f(a) = b_{0}. + \] + 任取 $a_{0}$ 满足 $f(a_{0}) = b_{0}$。 + 则有 + \[ + (g \circ f)(a_{0}) = c. + \] + \item 任取 $a_{1}, a_{2} \in A$。 + 假设 $f(a_{1}) = f(a_{2})$, 则 + \[ + g(f(a_{1})) = g(f(a_{2})), + \] + 即 + \[ + (g \circ f)(a_{1}) = (g \circ f)(a_{2}). + \] + 由于 $g \circ f$ 是单射, + \[ + a_{1} = a_{2}. + \] + \end{enumerate} \end{proof} %%%%%%%%%%%%%%% @@ -67,6 +125,11 @@ \end{problem} \begin{proof} + 首先, $f \circ g = I_{B}$ 是满射, 因此 $f$ 是满射。\\ + 其次, $g \circ f = I_{A}$ 是单射, 因此 $f$ 是单射。\\ + 所以, $f$ 是双射。\\ + 又由于 $f \circ g = I_{B}$, 因此 $g = f^{-1}$。 + \marginnote{该证明用到了课件中的两个定理, 请自行找出来。} \end{proof} %%%%%%%%%%%%%%% @@ -92,6 +155,45 @@ \end{problem} \begin{proof} + \begin{enumerate}[(1)] + \setcounter{enumi}{1} + \item 只需验证 + \begin{description} + \item[$\le$ 是自反的:] 任取 $Y \in X/\sim$。 + 设 $Y = [y]_{\sim}$。由于 $\preceq$ 是自反的, + \[ + y \preceq y. + \] + 根据 $\le$ 的定义, + \[ + [y]_{\sim} \le [y]_{\sim}. + \] + 即 + \[ + Y \le Y. + \] + \item[$\le$ 是反对称的:] + 任取 $Y \in X/\sim$, $Z \in X/\sim$。 + 设 $Y = [y]_{\sim}$, $Z = [z]_{\sim}$。 + \begin{align*} + &Y \le Z \land Z \le Y \\ + \implies & y \preceq z \land z \preceq y \\ + \implies & y \sim z \\ + \implies & [y]_{\sim} = [z]_{\sim} \\ + \implies & Y = Z. + \end{align*} + \item[$\le$ 是传递的:] + 任取 $W\in X/\sim, Y \in X/\sim$, $Z \in X/\sim$。 + 设 $W = [w]_{\sim}, Y = [y]_{\sim}$, $Z = [z]_{\sim}$。 + \begin{align*} + & W \le Y \land Y \le Z \\ + \implies & w \preceq y \land y \preceq z \\ + \implies & w \preceq z \\ + \implies & [w]_{\sim} \le [z]_{\sim} \\ + \implies & W \le Z. + \end{align*} + \end{description} + \end{enumerate} \end{proof} %%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%