名校
解题方法
1 . 设
,若非空集合
同时满足以下4个条件,则称
是“
无和划分”:
①
;
②
;
③
,且
中的最小元素大于
中的最小元素;
④
,必有
.
(1)若
,判断
是否是“
无和划分”,并说明理由.
(2)已知
是“
无和划分”(
).
①证明:对于任意
,都有
;
②若存在
,使得
,记
,证明:
中的所有奇数都属于
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a70b95c53fb6655721e2a8c61f5c2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8559db5cec89fb0ed29e8be8fdb0b1.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03710ecc47ca36cb01c337a71d300974.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b6e72a98cbc82cb24cb85aa3ab837f5.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35a2410ce34b36954ed4923e600d42f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
④
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e006283149b3d1662205b5271dd69db2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58f045d0c3275b992d4a4f90dcd20e63.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/408f3365f7c6767cd3f006022ee22413.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6da92a00c5e0121accc325e50f6492fe.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c8559db5cec89fb0ed29e8be8fdb0b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
①证明:对于任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efb6b675fa03f7268b8cbd1f1d91bd27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4003dc977c4cacda932927eed9c9d10.png)
②若存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8457b5be40500d437a83bb12e488b5eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09bd7ed301e00171b88549a8deb65035.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5203c10c41f8b8aaa4c9cc90f1f3271.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cffa35373ec4e4684107b42adb7a5161.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
您最近一年使用:0次
2024-06-10更新
|
134次组卷
|
2卷引用:北京市丰台区2023-2024学年高一上学期期末练习数学试卷
2 . 已知整数
,集合
,
,
,满足
,对任意的
,都有
且
.记
.
(1)若
,写出两组满足条件的集合
,
并写出相应的
;
(2)证明:
;
(3)求
的所有可能取值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/652fd852e080c91df90888d0449d63d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ea7fcdb5423c1c8c032a3efcf245682.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7bb58dca886fc65d874e2b30040c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f906607b8f6606742f496f73fff2ffd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/937c09d82c480e4d67f8a48d3f66c5f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/431acf301f0cf1e414b532de94708474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd31439e275791f4fcd3d22b5ae30ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ab4a9bfa50054c808dd8190305d0abd.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.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/ce4a2681390214200443ae07c01a4abe.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/607a272bff596f86f25dd99f5a69fbbe.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4a2681390214200443ae07c01a4abe.png)
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3 . 拓扑学是一个研究图形(或集合)整体结构和性质的一门几何学,以抽象而严谨的语言将几何与集合联系起来,富有直观和逻辑.已知平面
,定义对
,
,其度量(距离)
并称
为一度量平面.设
,
,称平面区域
为以
为心,
为半径的球形邻域.
(1)试用集合语言描述两个球形邻域的交集;
(2)证明:
中的任意两个球形邻域的交集是若干个球形邻域的并集;
(3)一个集合称作“开集”当且仅当其是一个无边界的点集.证明:
的一个子集是开集当且仅当其可被表示为若干个球形邻域的并集.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15e7cbf6370f2b5c37816278c4d52324.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cd50ba95ce394ae2cc7d8953268cad4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/528fd55bccdd48b002249e27153164dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84e93599300cd0cc2ee3747a0a1a01a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/331b36f89fa4fc1a314bd2fb469b6756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f853b9d71837401854312c2a3a2012d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/663819fd38d196961788cad4e2e039a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77c4e98464e40174ae21e741ae79dea5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/678bfef0c3cf7ee6438c64d20ab44617.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd934b73981f16a85a9a9d6554ec9791.png)
(1)试用集合语言描述两个球形邻域的交集;
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1985327691201a2fbcbb27689f2015.png)
(3)一个集合称作“开集”当且仅当其是一个无边界的点集.证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1985327691201a2fbcbb27689f2015.png)
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名校
解题方法
4 . 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)
您最近一年使用:0次
2023-09-17更新
|
361次组卷
|
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
5 . 已知
,
,
,记
,用
表示有限集合X的元素个数.
(1)若
,
,分别讨论
和
时,集合T的情况;
(2)若
,
,求
的最大值;
(3)若
,
,则对于任意的A,是否都存在T,使得
?说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1afd0680ce662524aa6451879d11808d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/952f1e0ce5bd53a6d5e8bb07ea2da5f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a65499f1d8f6e77e00e91d0f147ec6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dd5cb71dcc16e3321b8694d8394c12c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37765d2927d24d4b582423c843aebcd4.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a82f4f602933ea0b10f9eb8e63ce186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ca717c6a55e786238e64f7ebd69b9b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca31d722443aafe63d36132771c753c2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c345907ebe27888332b1b44c666cc47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a82f4f602933ea0b10f9eb8e63ce186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c32e5a6e7bf115636651cdfd3a6ec6ff.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764eff906937f9b1fb58e5abfb2eb8a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d197338ab810f6c9d31a2b67e5f352ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a82f4f602933ea0b10f9eb8e63ce186.png)
您最近一年使用:0次
6 . 已知集合
是集合
的子集,对于
,定义
.任取
的两个不同子集
,
,对任意
.
(1)判断
是否正确?并说明理由;
(2)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d39a39a70dacd0402f6fdd59a0e879.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc718cf6749da06b2dd90e9b2751854.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1171081d25910d6bb0bf9d32e82e33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d39a39a70dacd0402f6fdd59a0e879.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/0fc718cf6749da06b2dd90e9b2751854.png)
(1)判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51ac3d72ad811d426884f100f32aa0a2.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4090eff6726fa4e3a32655b3fba41abe.png)
您最近一年使用:0次
7 . 对于任意有限集
,定义集合
表示
的元素个数.已知集合
为实数集
的非空有限子集,设集合
.
(1)若
,求集合
和
;
(2)已知
为有限集,若
,证明:
.
(3)若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe7fd9b0c3c203a053a7ea52b71e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/185f1dec719b499d236ee7accaed0907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49c7673f4ca064bb1097f95523bf47cc.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab403f48a374c87fefc0c24923a063a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9281c61411eceeecf11c1f6ac31c2eec.png)
(2)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eccd49c9b9e3663880dac5b3029972a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c2198afa66c6a0cf4bb1698884da212.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09231ce23847f1780d130475ee341c96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27ddc772b27a6a72d3d6295f75e21298.png)
您最近一年使用:0次
2022-11-11更新
|
495次组卷
|
5卷引用:上海市行知中学2022-2023学年高一上学期期中数学试题
上海市行知中学2022-2023学年高一上学期期中数学试题北京市陈经纶中学2022-2023学年高一上学期12月诊断数学试题(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)期中真题必刷压轴30题-【满分全攻略】(沪教版2020必修第一册)(已下线)期中真题必刷压轴60题(15个考点专练)-【满分全攻略】(人教A版2019必修第一册)
8 . 已知集合
,
,
,且集合D满足
,
.
(1)求实数t的值:
(2)对集合
,其中
,定义由A中的元素构成两个相应的集合中:
,
,其中
是有序数对,集合S和T中的元素个数分别为m和n,若对任意的
,总有
,则称集合A具有性质P.
①请检验集合
与
是否具有性质P,并对其中具有性质P的集合,写出相应的集合S和T;
②试判断m和n的大小关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3010f543afc5dc2a5aa05b57733d6222.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c20bcf4b26b49a163924e15b1a481b7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a7edb1d3ea61fc8074dd93b96680430.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e5d8c14601bfbb14edcab1a040f118f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6678d814827717bafcb3d2c97a93fd3b.png)
(1)求实数t的值:
(2)对集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa3deeb60ea4e755eac7c7a4340e4c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d807832357bea22a266e63cbd7e678a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbafdbf1551f1189b7bee68792ee1f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/512a72ad5c5b298846ec842ec3ac0957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30277e0be448b4955903e81e8795e45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc020b0997a2f37b214718112b79d8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a2cacc52ffe015e828a4a5f2fe5ae4f.png)
①请检验集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/709a1f33195ab62f2da488b27a219c25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84601c1ff9db2761531f127cd020dd59.png)
②试判断m和n的大小关系,并证明你的结论.
您最近一年使用:0次
名校
解题方法
9 . 设集合
,
,
,
中至少有两个元素,且
满足:①对于任意
,若
,都有
;②对于任意
,若
,则
;
(1)判断下列两组集合是否满足要求:
(ⅰ)若
,则
;
(ⅱ)若
,则
;
(2)证明:若
有
个元素,则
有
个元素.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe7fd9b0c3c203a053a7ea52b71e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eef9096420672673840303a14f0fb636.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c6f228e37ac7282f2f013eda7395683.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe7fd9b0c3c203a053a7ea52b71e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fe7fd9b0c3c203a053a7ea52b71e7c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c36aecba41f6f5ff0d46a29dccaaf01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a11a069688e4c797fcf527eab15afa82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8339eab9c659e50db86828b65f825e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/566d386cbedb1c8750f4837633c2af64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a5abe56c019ac914e1fcde1865a747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5718e9c8baa106b447f9fae23e730de.png)
(1)判断下列两组集合是否满足要求:
(ⅰ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11e1540256cea69dcfb735c3e03eccdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d04d7b603e86497db23bc2b124a8e5c.png)
(ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38dbc6c0414c93f1dd3e4945bd34d082.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76bd82972fa678045162f19fee8142f.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af500a4e28d6f5b38390b7642eb96ed5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b06e95b57b7a81cd81d05557a11fa92.png)
您最近一年使用:0次
18-19高一上·北京·期中
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解题方法
10 . 给定数集A,若对于任意a,
,有
,
,则称集合A为闭集合.
(1)判断集合
,
是否为闭集合,并给出证明;
(2)若集合C,D为闭集合,则
是否一定为闭集合?请说明理由;
(3)若集合C,D为闭集合,且
,
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9df3a17aa370eba2add2c13cfc2619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7459675b810cf0b84696762ffc5c12f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4af08c31cb7887fad0ace3ad9fab61dd.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10b6c92b451ea80e63cbefd44c3681.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68e06607d8d217bce265fe228bd9401c.png)
(2)若集合C,D为闭集合,则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84601c1ff9db2761531f127cd020dd59.png)
(3)若集合C,D为闭集合,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf3ed15aa3dcc4211fb520b5b942c989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6857a6ef1ed2aeba64cd5c6dfd039dfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
您最近一年使用:0次
2022-08-28更新
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16卷引用:【全国百强校】北京市北京第四中学2018-2019学年高一上学期期中考试数学试题
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