1 . 设X,Y为任意集合,映射
.定义:对任意
,若
,则
,此时的
为单射.
(1)试在
上给出一个非单射的映射;
(2)证明:
是单射的充分必要条件是:给定任意其他集合
与映射
,若对任意
,有
,则
;
(3)证明:
是单射的充分必要条件是:存在映射
,使对任意
,有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86d7786f55f767e0a301b5032cdacc79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4645f4c284aef83b9d7edc909ce75168.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bd24e647a626899a243a3f3984f90a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f27822887caad20f3a075ca2fb74155c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
(1)试在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff048206370cd239052751e22f51089e.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d2e22788e03d6846442ce299b67bd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02ab73f0717678abfa9f8addc42be0c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce86cc928b228eba243c4bbfae8247ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683ee65c22bb8f6350feef3a3371a95f.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2216c6b89948de65d2e8488c2071a61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fd9b15b5cdf1c131ebf7cf2776cf7a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e0ab2f691659dfb9857e81cabc3142e.png)
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2 . 判断下列对应是不是从集合A到集合B的映射,其中哪些是一一映射?哪些是函数?为什么?
(1)A={1,2,3,4},B={3,4,5,6,7,8,9},对应关系f:x→2x+1;
(2)A={平面内的圆},B={平面内的矩形},对应关系是“作圆的内接矩形”;
(3)A={1,2,3,4},B={1,
,
,
},对应关系f:x→y=
.
(1)A={1,2,3,4},B={3,4,5,6,7,8,9},对应关系f:x→2x+1;
(2)A={平面内的圆},B={平面内的矩形},对应关系是“作圆的内接矩形”;
(3)A={1,2,3,4},B={1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf35027e76f8ea593f82023973d4aba3.png)
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2010·北京海淀·二模
名校
3 . 给定集合
,映射
满足:
①当
时,
;
②任取
若
,则有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
.
.则称映射
:
是一个“优映射”.例如:用表1表示的映射
:
是一个“优映射”.
表1 表2
(1)已知表2表示的映射
:
是一个优映射,请把表2补充完整(只需填出一个满足条件的映射);
(2)若映射
:
是“优映射”,且方程
的解恰有6个,则这样的“优映射”的个数是_____ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe05c3181726d202d021050807378989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92185b667af21e649c9b071042d29af0.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7873a6c263a103ce7a7b1410222bd245.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e2c865ebac120a4d1e17838ac9dc0ed.png)
②任取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea0eb2f420c6146a0809f01cb3168f49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72ac49ab7c8001c209b8611b9ea40d85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f1cd15d46d4fc519ea93da0a7a82a01.png)
.则称映射
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22c3373a475f821448143d482d7c0a35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea9eeefda7a3bfe23534700357fbacc.png)
表1 表2
![]() | 1 | 2 | 3 | |
![]() | 2 | 3 | 1 | |
![]() | 1 | 2 | 3 | 4 |
![]() | 3 |
(1)已知表2表示的映射
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fe7704aeb096266ab863326b66111d4.png)
(2)若映射
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf5648713989066021d4f7cd433727b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/987c640c84d230463ead6415c4c06a72.png)
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