(from Yanfei Dai) 设 $M$ 为自然数集, 试给出 $M$ 的两个双射变换 $\sigma,\tau$ 使得 $\sigma \tau\neq \tau\sigma$.
解答: 取 $$\beex \bea \sigma(1)=2,&\quad\sigma(2)=1,\\ \sigma(3)=4,&\quad\sigma(4)=3,\\ \sigma(5)=6,&\quad\sigma(6)=5,\\ \cdots\cdots,&\quad\cdots\cdots;\\ \tau(1)=3,&\quad\tau(2)=1,\\ \tau(3)=5,&\quad\tau(4)=2,\\ \tau(5)=7,&\quad\tau(6)=4,\\ \tau(7)=9,&\quad\tau(8)=6,\\ \cdots\cdots&\quad\cdots\cdots. \eea \eeex$$ 则 $\sigma$, $\tau$ 均为双射, 但 $$\bex \sigma\tau(1)=4\neq 1=\tau\sigma(1). \eex$$
