## Mathematics

Practical wisdom is only to be learned in the school of experience

# 20170911 | Exercise

1.若$P_1,P_2$为数域，证明：$P_1 \bigcap P_2$也为数域.

：设$P_1\bigcap P_2=P$，

$\because P_1,P_2$是数域，$\therefore 0,1\in P_1$，$0,1\in P_2$

$\therefore 0,1 \in P$

$\because P_1,P_2$是数域，$\therefore a+b\in P_1 , a+b \in P_2$

$\therefore a+b \in P$，$\therefore P$对加法运算是封闭的.

2.证明：集合$S=\left\{\dfrac{m}{2^n} | m,n\in Z\right\}$是一个数环.$S$是数域吗？

：令$m=n=0,\dfrac{m}{2^n}=0$，说明$S$非空；

$x\pm y=\dfrac{m}{2^n} \pm \dfrac{p}{2^q}=\dfrac{m \pm 2^{n-q} \cdot p}{2^n} \in S$

$xy=\dfrac{m}{2^n} \cdot \dfrac{p}{2^q}=\dfrac{mp}{2^{n+q}} \in S$

$\dfrac{x}{y}=\dfrac{\dfrac{m}{2^n}}{\dfrac{p}{2^q}}=\dfrac{\dfrac{m}{p}}{2^{n-p}}$，其中$\dfrac{m}{p}$不一定是整数，如$m=1,n=1,p=3,q=1,\dfrac{x}{y}=\dfrac{1}{3} \not\in S$

3.设$a,b \in \mathbf{R}$.证明：若对任何正数$\varepsilon$有$|a-b|<\varepsilon$，则$a=b$