名校
1 . 已知集合
为非空数集,定义:
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e092a46bd0eb089045217e1b13ccbb65.png)
(1)若集合
,直接写出集合
,
.
(2)若集合
,
,且
,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
,
,
,记
为集合
中元素的个数,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0d6adf2d45f344e1ca24c4cf0e4884c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e092a46bd0eb089045217e1b13ccbb65.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfa9c26b45f6947550f7c40cbfdecbba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0faaa2c9b8bc8ee534a517d6a108a66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3604274ad6707a906eba371a9e884144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76ad1e03a6ba59e8164e37c5e7e063e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b408aeda05ba8f05babea6251dcab2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a6a3d1be93cf6d16ee6e0ce0497f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650aa72fd4034517f9f981e382c1e50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b650aa72fd4034517f9f981e382c1e50.png)
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21卷引用:湖南省长沙市湖南师大附中2020-2021学年高一上学期第一次大练习数学试题
湖南省长沙市湖南师大附中2020-2021学年高一上学期第一次大练习数学试题北京大学附属中学2020-2021学年度高一10月考衔接班数学A层试题北京市海淀区中国人民大学附属中学2020-2021学年高一(上)期中数学试题北京市中国人民大学附属中学2020-2021学年度高一年级上学期数学期中练习试题北京人大附中2020-2021学年高一(上)期中数学试题(已下线)期末测试(能力提升)(1)-2020-2021学年高一数学(必修一)单元测试定心卷(沪教版2020)(已下线)高一上学期期末全真模拟02-2020-2021学年高一数学期末考试高分直通车(沪教版2020,必修一)(已下线)知识点03 交集、并集-2021-2022学年高一数学同步精品课堂讲+例+测(苏教版2019必修第一册)(已下线)第3课时 课后 集合的基本运算(已下线)第一单元 (基础过关)集合与常用逻辑用语 A卷-【双基双测】2021-2022学年高一数学同步单元AB卷(人教A版2019必修第一册)(已下线)第03讲 交集、并集(教师版)-【帮课堂】2021-2022学年高一数学同步精品讲义(苏教版2019必修第一册)第1章 集合(章末测试提高卷)-2021-2022学年高一数学同步单元测试定心卷(苏教版2019必修第一册)(已下线)专题03 集合的运算压轴题型-2021-2022学年高一《新题速递·数学》(人教A版2019)北京市广渠门中学2021-2022学年高一上学期期中质量检测数学试题(已下线)1.3 交集、并集(已下线)专题01 集合与逻辑(讲义)-2第一章 预备知识 期末综合复习测评卷-2021-2022学年高一上学期数学北师大版(2019)必修第一册(已下线)第3课时 课后 交集、并集(完成)(已下线)高一上学期期中【易错60题考点专练】(必修一前三章)-2022-2023学年高一数学考试满分全攻略(人教A版2019必修第一册)(已下线)第1章 集合与常用逻辑用语-【优化数学】单元测试基础卷(人教B版2019)(已下线)集合及其运算
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2 . 已知M是满足下列条件的集合:①
,
;②若
,则
;③若
且
,则
.
(1)判断
是否正确,说明理由;
(2)证明:
;
(3)证明:若
,则
且
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca05074e5a317ae45d073962bdf74dbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fefe237385a2dc1b005d8dc61ef56eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d57c23eb9af964e65a91af487f4c66cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c443466385f21cd3f06e2e4229add79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acfc595518cf752e1c7903dfff93dbda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38f0e9c04402a0ffdaa25c3e3c82c7dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8ac8248bb70f9ef5b0cb7d025e05160.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bee824cf4b0d0011adabfec2ef759d1c.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3143beb0df5feff68bf6dd94fdfd6b5b.png)
(3)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d57c23eb9af964e65a91af487f4c66cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c4dffdf9c06c0d11d0410f194afc8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8f06a4a9deb51418c20e7e7376cc807.png)
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4卷引用:江苏省南通市通州高级中学2020-2021学年高一上学期第一次阶段性测试数学试题
3 . 对于由m个正整数构成的有限集
,记
,特别规定
,若集合M满足:对任意的正整数
,都存在集合M的两个子集A、B,使得
成立,则称集合M为“满集”,
(1)分别判断集合
与
是否为“满集”,请说明理由;
(2)若
由小到大能排列成公差为d(
)的等差数列,求证:集合M为“满集”的必要条件是![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14f5fdf0e4f9de36f08402dd96d237e.png)
或2;
(3)若
由小到大能排列成首项为1,公比为2的等比数列,求证:集合M是“满集”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e495e34870eb6eef8486f88e567c7e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8b9cd59e58555bfc92257ba31d16794.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44aed8cc107aecae26873891bfdc5f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf084cdb896062c63e919adf38352d26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c01a90899f08d43e7f1b945b96aae753.png)
(1)分别判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4804e8c356d6aa5b0d645fed77fec88f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee5d150c4bf3836b14db9cd1017aeacd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20655342f9ace8b50a50f5eae6f37beb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e789387bf82c893b83cb8f2007f060.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14f5fdf0e4f9de36f08402dd96d237e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29f0913464ddee73888f859ec6ad1696.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20655342f9ace8b50a50f5eae6f37beb.png)
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4卷引用:北京市人大附中朝阳学校2020-2021学年高二下学期数学统测试题
北京市人大附中朝阳学校2020-2021学年高二下学期数学统测试题上海市松江区2021届高三上学期期末(一模)数学试题上海市松江区2021届高三高考数学一模试题(已下线)考点47 推理与证明-备战2022年高考数学(文)一轮复习考点帮
名校
4 . 已知集合
.对于
,定义:
与
的差为
;
与
之间的距离为
.
(1)当
时,设
,求
;
(2)若对于任意的
,有
,求
的值并证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4064df6eecf5283706a8ab2abbddad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a399c4f94df181a7527ff028b3b40ab3.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/becab20e5cae4a3ad36eb9be7e1a5e3a.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/badd0969b43deabb1e8f3fcca73ce1c5.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a59b86d73479a6550e86eadb20b39d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67cf5b4308f0a73d645704d1dfe9b9ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3c11e3798218ef9e0c49b1362403f4f.png)
(2)若对于任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81345ca73b711411e665820b5672913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42a0663324985d2901305e58b066e763.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8513f18376e4e456b939d0f1cdb6e602.png)
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5卷引用:河南省新乡市原阳县第一高级中学2023-2024学年高一上学期1月月考数学试题
河南省新乡市原阳县第一高级中学2023-2024学年高一上学期1月月考数学试题北京市昌平区2020-2021学年高一上学期期末质量抽测数学试题(已下线)第一章 集合与常用逻辑用语(选拔卷)-【单元测试】2021-2022学年高一数学尖子生选拔卷(人教A版2019必修第一册)北京市延庆区2021-2022学年高二下学期期末数学试题北京市北京师范大学附属实验中学2022-2023学年高一上学期期中考试数学试题
名校
5 . 设A,B为两个集合,我们定义集合
为两个集合A,B的差集,记为A-B
(1)已知
,求
和
.
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8054bd86268fb956685c0883f135f406.png)
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59d46cde3e85dfc5b9c71f53265fb03a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41604a15c2ec813ad37bd0a9bfa33136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7845a8c978192626a74e63f6c8bd8e9.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663ff7fb37bdd6c89a2ab93904fd95b2.png)
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6 . 已知集合
为非空数集,定
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d0b31a3a2f0e3d81036338bc74c0bd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b70419d498a0672197837b69bb5811.png)
(1)若集合
,直接写出集合
及
;
(2)若集合
,
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/059d5b21057e68e84dae9af30b86cd09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d0b31a3a2f0e3d81036338bc74c0bd6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b70419d498a0672197837b69bb5811.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/265e81e52218232e16c78f57b3aa0de9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b26ea9b3e6e286874c5dca1badea723.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e16289945d1d1c529fb1bfd4d828f413.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7591668ec1bc5e8b1f2c4dba4835d1b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3604274ad6707a906eba371a9e884144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9473a5974fa9c4286f90f6a3637411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
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7 . 记有理数集
的非空子集
具有以下性质:①
;②若
,
,则
;③存在非零有理数
,
且每一个不在
中的非零有理数都可写成
的形式,其中
.
(1)若
,
,求证:
;
(2)若
是非零有理数,且
,求证:
;
(3)求证:
,则存在
、
,使
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e680c82535270feea54ec5cc81fcc99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d05f1347d2bcfdd009cd92b4528db70d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2bd6719b88b02f805cd280a273317686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c99a6c6f64340823bb8cd67360cff9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72c70077b84eda1b47e2acae9fd5bb8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aa8a716a31b0f51b70fdf9bdb257909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e337060df2f24d9d83cd33c1d3f1322.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9aafde6d7fe0d22936f82c23ad944dfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084df0babb739a54a118e5a1b6800891.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/084df0babb739a54a118e5a1b6800891.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/013ee61d98c06bf3601bf752676d2e26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa2a702b01ea4b4cadb6b74024daaed0.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad481cbfb67ac9cdbc0537f3de23b022.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15106af5bc32a503a164d7d183f7cdaa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f41a08c160ca2373f99628d5d2125c90.png)
(3)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c8647b00cc8c8f35555c7d78cf2812c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bfc4d1adffa8e29873658f8ed62a9ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/171f0e6da3d34eb9b71152ae5b569d66.png)
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名校
8 . 已知集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1edda0a840fa9fd056a84b3851fe2840.png)
,其中
,定义
.若
,则称
与
正交
(1)若
,写出
中与
正交的所有元素
(2)令
.若
,证明:
为偶数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1edda0a840fa9fd056a84b3851fe2840.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc70211f57c2993d7047c89fd3fc2e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb56545f5200342abd9e406cb200328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9117bd5df08f7f5cf00d1bd3ced32e0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4157967918cabbed7f5d82a291cc262f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b623d5f3c82cfdc60c0d74f381e6e1f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e47cd514b2920609e3781c87df6ab70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)令
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c4bd27eda92741b7923451594321771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22111b1f07e7873e5a156d1937eaac27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fd1f0ace9ca0b79929e73af6c201c2e.png)
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9 . 设集合
满足条件,若
,则
(
且
).
(1)若
,求集合
;
(2)若
,试证明:
;
(3)集合
能否为单元素集合?为什么?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74137fa46f6f3208f5924cb1b8c66b08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24a57996290794e082b21d8f1dfc322a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e30c903d8f8a05332af0b19e7e40df3.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed961de27af72b7d11887ccfb6f15071.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73d59d461091895f698c5d33dcb9c804.png)
(3)集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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名校
10 . 设
为正整数,集合
.对于集合
中的任意元素
和
,记![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26afd72eb1015c1c57136d8839a058a6.png)
(1)当
时,若
,
,求
和
的值
(2)当
时,对于
中的任意两个不同的元素
,
.证明:
.并举一个使得等号成立的
,
的例子.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da48fd6c3372edfdaa91396a6a939d34.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8c7341e625d7c3874599342228e5ea1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e37fb10de30ed5690cdc8c069547d6ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26afd72eb1015c1c57136d8839a058a6.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3ac2a84fb2a60ae15d7fb3f3b910a7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa9ba4ced7197552108e07a108508702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5dc2e141aa34f5340459fc6c0214cb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9af33c0896c2f23fdd2e2537c68d35da.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90a3060d2dfc450c6b3f06baaf2df71e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
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