1 . 设数集
满足:①任意
,有
;②任意
、
,有
或
,则称数集
具有性质
.
(1)判断数集
是否具有性质
,并说明理由;
(2)若数集
且
具有性质
.
(i)当
时,求证:
、
、
、
是等差数列;
(ii)当
、
、
、
不是等差数列时,写出
的最大值.(结论不需要证明)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8647b00cc8c8f35555c7d78cf2812c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6e2e79843faf62dde86bf858d1e0569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c8b2714e2f6ddfdd6b05d3b4de1149.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b8e5872f45d4b878b0119997cb5bae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84734fbba70c0b45045fabf8090f810b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(1)判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2161a642b95463642adc3892850bc74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)若数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ddc189d2c675b0e2ade4f7ed40f66fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8509c4b8fef1e10a20fe1c3e9243ac8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(i)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76c7867969b14fd642147188b6ebf29c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(ii)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e72adb45c60c2f63b46e65ff787302bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e88093a749c0d46e0ee931ecfaff925.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa5e9bd516f6282483b92cfe6074623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
2021-09-26更新
|
589次组卷
|
7卷引用:重难点08 七种数列数学思想方法-2
解题方法
2 . 我们知道,二维空间(平面)向量可用二元有序数组
表示;三维空间向盘可用三元有序数组
表示.一般地,
维空间向量用
元有序数组
表示,其中
称为空间向量的第
个分量,
为这个分量的下标.对于
维空间向量
,定义集合
.记
的元素的个数为
(约定空集的元素个数为0).
(1)若空间向量
,求
及
;
(2)对于空间向量
.若
,求证:
,若
,则
;
(3)若空间向量
的坐标满足
,当
时,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1acd1459dd96e861e6e04abccb2a3817.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c2a29087dbd2e7635da13f7d288c1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5a3c6c6fe94124d76957d9a8c837701.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8715a3f984d2627afd7c40c61347b7cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc4e5fb9e41d1310778b0dda692066dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afad70ea217c830631639e8508ad410b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0196cb738ee760339a9e15c8e6d9a41.png)
(1)若空间向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd4517944d03f267b87ee1c184f463dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7a0d7cdd1e3a38753d1290d9de9f9af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecc51f796615bfd474cee9d4d80e1eae.png)
(2)对于空间向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/086eb439f6a1578fdba904825340772d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09611387d4de23004d388c9a8dde3438.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daee8af73118698c77e022651f69ef22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1a205f096c854a2f7cd71255056f9f7.png)
(3)若空间向量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bd074bea5f4eb8f60729b75e970afda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3894099d6bf29b73086842a48da10174.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1653bed33a88140f16d494e8454f5225.png)
您最近一年使用:0次
3 . 给定整数
,由
元实数集合
定义其随影数集
.若
,则称集合
为一个
元理想数集,并定义
的理数
为其中所有元素的绝对值之和.
(1)分别判断集合
是不是理想数集;(结论不要求说明理由)
(2)任取一个5元理想数集
,求证:
;
(3)当
取遍所有2024元理想数集时,求理数
的最小值.
注:由
个实数组成的集合叫做
元实数集合,
分别表示数集
中的最大数与最小数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a74fd362793540afbf0d97d96e34f86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19a004e53d6da57977a26325a5336e6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798fbe13094bfa183eb57fae77647508.png)
(2)任取一个5元理想数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b942b7722c385c79c70ba8928d38af.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd2861d53cafb06f076c59b25bdfd857.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
注:由
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcc0cbd6a65ad74e65716c682d6985cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
您最近一年使用:0次
4 . 已知集合
,对于
,
,定义
与
之间的距离为
.
(1)已知
,写出所有的
,使得
;
(2)已知
,若
,并且
,求
的最大值;
(3)设集合
,
中有
个元素,若
中任意两个元素间的距离的最小值为
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46717c433c9f34b936f694eba762ccb0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d0135cb7ffda469b422df0aa1817d90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c76f1774902da4c8da18791b5a35c65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80ebc8c7e32c1b561a908a36cfa2cbb5.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3231e2c7d20174642764c2a96427636d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c72cbaca91d8e578762c4f0b6750a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e79b6edfd5f5bb3866599fcb2aa6f3.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea7b72192ddc13754ebc4c3d70b40ea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e77ba8c90d21237670483bbcd8ac63b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8b9dbb9c52862ac974003241d68fb61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3bcb4828b16c8e845492f1a53ddd9a9.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a2b280a6a65c9bf2f20cad5d1106080.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2ea5126ca30d68cc8b70f03860ba2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36a1b09c653185842513e24ebba60bb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9d528f49ab8d7fd971c5bb8fa7f24d4.png)
您最近一年使用:0次
名校
解题方法
5 . 设k是正整数,A是
的非空子集(至少有两个元素),如果对于A中的任意两个元素x,y,都有
,则称A具有性质
.
(1)试判断集合
和
是否具有性质
?并说明理由.
(2)若
.证明:A不可能具有性质
.
(3)若
且A具有性质
和
.求A中元素个数的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/858911660b233271d57b17e358232d45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3cf0ebf259b9007acfffe8b6940abc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46bf6ded2f869744c6c50785f974aa6.png)
(1)试判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d167be863d109213bd07becd62b74d12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c73a7a7e9ecb2c8296e505e5409fb2ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bea0dd7e474bcd04db2544427ba0488.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d11d851264c4ef68ea96f895c0136d0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7470297de40027847c5c73fc5d1719c.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bacd2a1627ef91a38a03ac4e32adc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c414a10d73f453fc1109e5b2243d2369.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb1832cb6b4e96e3d4f34d79b0e88854.png)
您最近一年使用:0次
名校
6 . 已知
是各项均为正整数的无穷递增数列,对于
,定义集合
,设
为集合
中的元素个数,若
时,规定
.
(1)若
,写出
及
的值;
(2)若数列
是等差数列,求数列
的通项公式;
(3)设集合
,求证:
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/542b4acf7b25b750fbe7205fd179b978.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/857369257ea1b23ef40ce7e3a0f058af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/233427826eb2233641fc3a9805f6d206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1202d58cd3ad66e7b23f01024566705b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1cc57d8a4f67a040435d8b206d3254bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6510d0816033afa001c130342bb7cda.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e4b5779873cb3f4366dbfdb983dec81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b6f99a33b14f53fb398a195aa2ec3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac648580405ecaa29e91d45738a08af7.png)
(2)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce83115a50f99e08e9a2db7267aeed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63d471926f7b27322d90c82b9ce21d3d.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b54e4701d4cb8d0133ad2044a7e0f52e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1479e28bf6a8cb64ec7df77cd295f99d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a6a3d1be93cf6d16ee6e0ce0497f46.png)
您最近一年使用:0次
2024-01-21更新
|
1353次组卷
|
7卷引用:专题1 集合新定义题(九省联考第19题模式)讲
(已下线)专题1 集合新定义题(九省联考第19题模式)讲(已下线)2024年高考数学二轮复习测试卷(北京专用)(已下线)黄金卷01(2024新题型)(已下线)微考点4-1 新高考新试卷结构压轴题新定义数列试题分类汇编北京市朝阳区2024届高三上学期期末数学试题广东省江门市开平市忠源纪念中学2024届高三下学期高考冲刺考试(一)数学试卷江苏省常州市华罗庚中学2024届高三下学期4月二模训练数学试卷
7 . 已知点集
满足
,
,
.对于任意点集
,若其非空子集A,B满足
,
,则称集合对
为
的一个优划分.对任意点集
及其优划分
,记A中所有点的横坐标之和为
,B中所有点的纵坐标之和为
.
(1)写出
的一个优划分
,使其满足
;
(2)对于任意点集
,求证:存在
的一个优划分
,满足
;
(3)对于任意点集
,求证:存在
的一个优划分
,满足
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d82ea5bd5d061a16d0ba32daacb6dece.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0efaead32620574f09623dff495df983.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4de122ae929b1acaff321dec137622ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9610877bf323ad7e27b83e886220f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dea9a4259cca10c1f5af28e621ebafd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fc0f5296284e2f6dcba4dbb77b0e141.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12cb68674b3cdaeda8b7123c747460fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a47112c5ff4179fca6e28aa4d0d47d21.png)
(1)写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16b65432b8fcd588460df55bb8a21f65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/673bf8d12ff9956f1fe4b31d32c201fd.png)
(2)对于任意点集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32efe4eff75508cb93e828c735dcb695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32efe4eff75508cb93e828c735dcb695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16fd919250fa4a36bc880e8038cbc90b.png)
(3)对于任意点集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ddad3d9fdb5e9951b6a1c31f9a72a71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/883f136eeb68b72dc768c697ae31e535.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9505c141fd8b7c6b4139667bb5853648.png)
您最近一年使用:0次
名校
8 . 定义1:对于一个数集
,定义一种运算
,对任意
都有
,则称集合
关于运算
是封闭的(例如:自然数集
对于加法运算是封闭的).
定义2:对于一个数集
,若存在一个元素
,使得任意
,满足
,则称
为集合
中的零元,若存在一个元素
,使得任意
,满足
,则称
为集合
中的单位元(例如:0和1分别为自然数集
中的零元和单位元).
定义3:对于一个数集
,如果满足下列关系:
①有零元和单位元;
②关于加、减、乘、除(除数不为0)四种运算都是封闭的;
③对于乘法和加法都满足交换律和结合律,且满足乘法对加法的分配律,则称这个数集
是一个数域.
(1)指出常用数集
中,那些数集可以构成数域(不需要证明);
(2)已知集合
,证明:集合
关于乘法运算是封闭的;
(3)已知集合
,证明:集合
是一个数域.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57e35231b964e293122c4383dac2431b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bfad095b65beb6e77aee3664b0e6bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad135b14c9dcd83eab6618d7694c7b0.png)
定义2:对于一个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc020b0997a2f37b214718112b79d8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac385ec112e6d61b90d953e3f106ee85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29df165b5fc74dcbc39df74d541148da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9df3a17aa370eba2add2c13cfc2619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac385ec112e6d61b90d953e3f106ee85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c4e0c03c5aef711627f1b3124d8b5fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad135b14c9dcd83eab6618d7694c7b0.png)
定义3:对于一个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
①有零元和单位元;
②关于加、减、乘、除(除数不为0)四种运算都是封闭的;
③对于乘法和加法都满足交换律和结合律,且满足乘法对加法的分配律,则称这个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)指出常用数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39021e0f386119bd1b5c73b843106e55.png)
(2)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37b491b231d0ff8a387813b5686579f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4c1c450038724991d6d39ecefae405.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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9 . 定义1:通常我们把一个以集合作为元素的集合称为族(collection).
定义2:集合
上的一个拓扑(topology)乃是
的子集为元素的一个族
,它满足以下条件:(1)
和
在
中;(2)
的任意子集的元素的并在
中;(3)
的任意有限子集的元素的交在
中.
(1)族
,族
,判断族
与族
是否为集合
的拓扑;
(2)设有限集
为全集
(i)证明:
;
(ii)族
为集合
上的一个拓扑,证明:由族
所有元素的补集构成的族
为集合
上的一个拓扑.
定义2:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a837165ca03f9e4ea8964979c95e3bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(1)族
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ee5abf02995c6ac2135347a663cdb0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e92ee8109b0f949f8946814f0a69e8b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(2)设有限集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(i)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a75d83dc31194727f441b79eee9cfc.png)
(ii)族
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9350d17e3ce2d85030a0076b53174a96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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10 . 已知集合
中含有三个元素
,同时满足①
;②
;③
为偶数,那么称集合
具有性质
.已知集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2472afe9a850dd4ed6e2d5f0dc986ec9.png)
,对于集合
的非空子集
,若
中存在三个互不相同的元素
,使得
均属于
,则称集合
是集合
的“期待子集”.
(1)试判断集合
是否具有性质
,并说明理由;
(2)若集合
具有性质
,证明:集合
是集合
的“期待子集”;
(3)证明:集合
具有性质
的充要条件是集合
是集合
的“期待子集”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a14c388e1e2e5a2ff1ccf6caffbee0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d561cdf17d3a6d7d57af719cb6100c85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55de080a3c1313bbd6ea5c17307c723.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1433c8103033c67232f2f9ae189608d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2472afe9a850dd4ed6e2d5f0dc986ec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bccfdca384d27d0d8395068e21770157.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b60809c809060f8d3f7a2d13b4baf6cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
(1)试判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f640ba418f74a232666dc3fb3abfaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ff95c388d22eba3d62c196277b47a79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f30f56664446f32dbbc2c5f12a99374.png)
(3)证明:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
您最近一年使用:0次
2024-03-07更新
|
1882次组卷
|
4卷引用:专题1 集合新定义题(九省联考第19题模式)讲
(已下线)专题1 集合新定义题(九省联考第19题模式)讲(已下线)专题1 集合新定义题(九省联考第19题模式)练广东省东莞中学、广州二中、惠州一中、深圳实验、珠海一中、中山纪念中学2024届高三第四次六校联考数学试题安徽省芜湖市安徽师大附中2023-2024学年高二下学期3月测试数学试题