若集合P={x|2x-a<0},Q={x|3x-b>0},a,b∈N,且P∩Q∩N={1},则满足条件的整数对(a,b)的个数为 .
【答案】
分析:由集合P={x|x<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/0.png)
},Q={x|x>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/1.png)
},得 P∩Q={x|
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/2.png)
>x>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/3.png)
},由P∩Q∩N={1},a,b∈N,
可得1<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/4.png)
≤2,1>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/5.png)
≥0,故 a=3或4,b=0,1,2.
解答:解:∵集合P={x|2x-a<0}={x|x<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/6.png)
},Q={x|3x-b>0 }={x|x>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/7.png)
},a,b∈N,且P∩Q∩N={1},
∴P∩Q={x|
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/8.png)
>x>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/9.png)
},
∴1<
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/10.png)
≤2,1>
![](http://thumb.1010pic.com/pic6/res/gzsx/web/STSource/20131101222049950130812/SYS201311012220499501308009_DA/11.png)
≥0,∴2<a≤4,0≤b<3,∴a=3或4,b=0,1,2,
故满足条件的整数对(a,b)的个数为6,
故答案为6.
点评:本题考查集合的表示方法,两个集合的交集的定义和求法,解不等式,求得a=3或4,b=0,1,2,是解题的关键.