名校
1 . 已知集合
为非空数集,定义
.
(1)若集合
,请证明
,并直接写出集合
;
(2)若
且
,集合
,求
的最小值;
(3)若集合
,且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a1218ca84c0ea386cc4af4a7d25fb7d.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fd6edb659be68495364855860dca3ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3e7fd6bf379008c85f6cf6f85871a60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e16289945d1d1c529fb1bfd4d828f413.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98ca33894cfd022eb3a57cfde78f06b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd4613271f782a90ab580131d09d03d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3489591aa8bf18d0c4c4363964c234db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77239c98c78a026cc03336edca067ea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9473a5974fa9c4286f90f6a3637411.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
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2 . 已知集合
为非空数集,定义:
,
(实数a,b可以相同)
(1)若集合
,直接写出集合S、T;
(2)若集合
,
,且
,求证:
;
(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/b72b157bd4a8e6ef02c906a5bd99bad2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7b9122f884314b4fb300d800d82cbd2.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceb842937a0a1269188e84331a909afd.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/c76ad1e03a6ba59e8164e37c5e7e063e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ae30071e46588524faa5c6f8a65aa26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a6a3d1be93cf6d16ee6e0ce0497f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
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名校
3 . 集合A为非空数集,定义:
,
.
(1)若集合
,直接写出集合S、T;
(2)若集合
,
,且
,求证:
;
(3)若集合
,
,记
为集合A中元素的个数,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b47cb8e325f4e0ded238fcd80b5ce954.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1abd6a401084df3b4f8661000db4307f.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a22f8d9188279429b395c30b2898ac1.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/c76ad1e03a6ba59e8164e37c5e7e063e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/582a9db63ddc5da3571ade7d0958f6f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30a6a3d1be93cf6d16ee6e0ce0497f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
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2023高一·全国·专题练习
4 . (1)已知集合
且
,任意从中取出k个四元子集
,均满足
的元素个数不超过2个,求k的最大值.(举出一个例子即可,无需证明)
(2)已知集合
且
,任意从中取出k个三元子集
,均满足
的元素个数不超过一个,求k的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74cbafbbc7f78431a6f041f87844178c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43e4a73e0269c6b90c9c620a27119db5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9d48660f696fc51b97c89fa55d6f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738aa208e410305523fa64b8518ba6b2.png)
(2)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f04bf835c1c32d9e61922191da7848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43e4a73e0269c6b90c9c620a27119db5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc9d48660f696fc51b97c89fa55d6f4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/738aa208e410305523fa64b8518ba6b2.png)
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5 . 设
是非空数集,若对任意
,都有
、
,则称
具有性质
,给出以下命题:
①若
具有性质
,则
可以是有限集;
②若
具有性质
,且
,则
具有性质
;
③若
、
具有性质
,且
,则
具有性质
;
④若
、
具有性质
,则
具有性质
.
其中所有真命题的序号是______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e13a814f8e081078dcf3788177affcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ae4f0ccdfc1206d809e581449d0452e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
①若
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
②若
![](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/f20c225a51cdde922ccae6bd01d970cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c6009ab1bad0e053bc7e11b868cac6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
③若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a8543bb9be52b25cf5be0a39110c9e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b76d26b78e63683dfacf10d3da6d74d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
④若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2646f41226f24960a6186dc7860ef45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
其中所有真命题的序号是
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名校
解题方法
6 . 若X是一个非空集合,M是一个以X的某些子集为元素的集合,且满足:①
,
;②对于X的任意子集A,B,当
且
时,有
;③对于X的任意子集A,B,当
且
时,有
,则称M是集合X的一个“M-集合类”.例如:
是集合
得一个“M—集合类”.若
,则所有含
的“M—集合类”的个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cbe57631a9011daf7b64de2183caefb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/404a8c04ce2b2bdd59d2e9b4eca437a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc3068bce2d48b79353bb8dc65260bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74f80fced6fa7e6adc72c80228443885.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b675c905836864759637f98ae27c778c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7fc3068bce2d48b79353bb8dc65260bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74f80fced6fa7e6adc72c80228443885.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ffcc39b31b83ff0ea33a77a4e5b671b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4a7274d8d531b3a39ff8a70b1b6c591.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fc4fd4aae1c3be924a1cdf07f54ac26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b5c073137c7a5dc32c9376667417a26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46c7e50f57c88aa5d10676c5e16e4756.png)
A.9 | B.10 | C.11 | D.12 |
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2023-10-13更新
|
319次组卷
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7卷引用:湖南省岳阳市2022-2023学年高一下学期期中数学试题
湖南省岳阳市2022-2023学年高一下学期期中数学试题(已下线)高一上学期第一次月考选择题压轴题50题专练-举一反三系列山东省烟台市招远市第二中学2023-2024学年高一上学期10月月考数学试题(已下线)专题01 集合及其运算2-【寒假自学课】(苏教版2019)(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)上海市实验学校2015-2016学年高一上学期期中数学试题(已下线)专题02 集合与常用逻辑用语常考压轴题型-2021-2022学年高一《新题速递·数学》(人教A版2019)
7 . 19世纪戴德金利用他提出的分割理论,从对有理数集的分割精确地给出了实数的定义,并且该定义作为现代数学实数理论的基础之一可以推出实数理论中的六大基本定理.若集合A、B满足:
,则称
为
的二划分,例如
,
,则
就是
的一个二划分,则下列说法正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea98d5089f9c44d87ef8c235ef22b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62d40a270d7c68c25d5729746dab549f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292378edf9103ac1391f172534927b1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3c351231c4ecf4f8904b4ca2a7b1969.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94b4adcb6d93738c01d5fbec0d649af9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62d40a270d7c68c25d5729746dab549f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/292378edf9103ac1391f172534927b1f.png)
A.设![]() ![]() ![]() |
B.设![]() ![]() ![]() |
C.存在一个![]() ![]() ![]() ![]() |
D.存在一个![]() ![]() ![]() ![]() ![]() ![]() |
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2023-09-26更新
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11卷引用:浙江省宁波市北仑中学2022-2023学年高一下学期期初返校考试数学试题
浙江省宁波市北仑中学2022-2023学年高一下学期期初返校考试数学试题(已下线)高一上学期第一次月考数学试卷(提高篇)-举一反三系列江西省宜春市宜丰县宜丰中学创新部2022-2023学年高一下学期第一次月考数学试题重庆市缙云教育联盟2023-2024学年高一上学期9月月度质量检测数学试题辽宁省沈阳市第二中学2023-2024学年高一上学期9月月考数学试题重庆市松树桥中学校2023-2024学年高一上学期第一次月考数学试题(已下线)单元高难问题01集合中的新定义问题-【倍速学习法】(人教A版2019必修第一册)(已下线)第一章 集合与常用逻辑用语(压轴题专练)-速记·巧练(人教A版2019必修第一册)(已下线)高一上学期期末考试选择题压轴题50题专练-举一反三系列浙江省杭州第二中学2022-2023学年高一上学期期中数学试题(已下线)高中数学-高一上-56
名校
解题方法
8 . 已知集合
为非空数集,定义:
,
(1)若集合
,直接写出集合
(无需写计算过程);
(2)若集合
,且
,求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb1868c8a0db983c9cc2695294fa03b1.png)
(3)若集合
,记
为集合
中的元素个数,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/312ab0127457545c7bac2193505e0a6b.png)
(1)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16e8d171c3d2d86dc5594cffb51096fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0a44fa352255b81702d306cb32cf468.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77239c98c78a026cc03336edca067ea3.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/df87128f2e2850f12dc88216def10cc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc8279d9dd0b7750953cb9e2098b3b90.png)
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2023-09-17更新
|
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3卷引用:上海市高桥中学2023-2024学年高一上学期月考(一)数学试题
上海市高桥中学2023-2024学年高一上学期月考(一)数学试题(已下线)高一上学期期中考试解答题压轴题50题专练-举一反三系列北京市中央民族大学附属中学(朝阳)2023-2024学年高一上学期期中考试数学试题
名校
9 . 设m是实数,已知集合
,集合
,且
,则m的取值范围是_______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc088345fb1d4cfdda89d068dd0f93d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/104668ced17a26db2ae1862b9f0b23c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e63128a77392afc5f8b873b7a67f2b5f.png)
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2023-09-11更新
|
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4卷引用:上海市敬业中学2024届高三上学期开学考试数学试题
上海市敬业中学2024届高三上学期开学考试数学试题海南省海口市琼山华侨中学2023-2024学年高二上学期期中考试数学试题海南省省直辖县级行政单位澄迈县澄迈中学2023-2024学年高二上学期期中数学试题(已下线)2.1.4 圆与圆的位置关系(十大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020选择性必修第一册)
10 . 对于正整数集合
,
,如果去掉其中任意一个元素
之后,剩余的所有元素组成的集合能分为两个交集为空集的集合,且这两个集合的所有元素之和相等,我们就称集合
为“和谐集”
(1)判断集合
是否是“和谐集”,并说明理由.
(2)判断集合
是否是“和谐集”,并说明理由.
(3)求证:集合
不是和谐集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8a2094e3909dbce5d966776a5cb847.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2613279bffd089060f0d05e48eabd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42c015b7ebebf921e559369b98bc98d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0914b68f106a912420705b2f3928ca42.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcb62f10c9971d5aafff76dc4dfb4732.png)
(2)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b91f2f906face10cd95d22d83921abc.png)
(3)求证:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032b0be0be95b689ebcf3ccdfd059652.png)
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