题目内容
设※是集合A中元素的一种运算,如果对于任意的x,y∈A都有x※y∈A,则称运算※对集合A是封闭的,若M={x|x=a+![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_ST/0.png)
A.加法
B.减法
C.乘法
D.除法
【答案】分析:这是一个简单的合情推理问题,根据新运算的定义:果对于任意的x,y∈A都有x※y∈A,则称运算※对集合A是封闭的,要判断M={x|x=a+
b,a,b∈Z}时,则对集合M不封闭的运算,我们可以对答案中对应的运算逐一进行判断,易得到答案.
解答:解:设x=a+
b,y=c+
d(a、b、c、d∈Z),
则x+y=(a+c)+
(b+d)∈M,所以加法对集合M封闭;
x-y=(a-c)+
(b-d)∈M,∴减法对集合M封闭.
xy=(ac+2bd)+
•(ad+bc)∈M,∴乘法对集合M封闭.
=
=![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/8.png)
=
+![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/10.png)
,
故除法对集合M不封闭,
故答案选D.
点评:这是一道新运算类的题目,其特点一般是“新”而不“难”,处理的方法一般为:根据新运算的定义,将答案中的运算逐一代入进行验证,易得最终结果.
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/0.png)
解答:解:设x=a+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/1.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/2.png)
则x+y=(a+c)+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/3.png)
x-y=(a-c)+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/4.png)
xy=(ac+2bd)+
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/5.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/6.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/7.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/8.png)
=
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/9.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/10.png)
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131103100020169872707/SYS201311031000201698727002_DA/11.png)
故除法对集合M不封闭,
故答案选D.
点评:这是一道新运算类的题目,其特点一般是“新”而不“难”,处理的方法一般为:根据新运算的定义,将答案中的运算逐一代入进行验证,易得最终结果.
![](http://thumb2018.1010pic.com/images/loading.gif)
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