名校
解题方法
1 . 若集合
中恰有
个元素,则称函数
是“
阶准偶函数”.已知函数
是“2阶准偶函数”,则
的取值范围是________
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d49387f7cd88490ab8c48ea163d4af37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/309c33435e66a747cda2967ccea6d6bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
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3卷引用:考点3 与集合相关的新定义问题 --2024届高考数学考点总动员【练】
(已下线)考点3 与集合相关的新定义问题 --2024届高考数学考点总动员【练】安徽省安庆市第一中学2023-2024学年高一上学期12月“三新”检测考试数学试题云南省昭通市教研联盟2023-2024学年高一上学期期末考试数学试卷
名校
解题方法
2 . 已知集合
中含有
个元素,集合
是
的非空子集,且
,则不同的集合对
有______ 个.(用含
的代数式表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b52b4f24969673c863b5aff4fb6751ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4f27f84764f1cca89ce3d93fc1cf603.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01c74a907dda6bb7d9d56d009d9df253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b52b4f24969673c863b5aff4fb6751ce.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/a95d247ffb2774bb7e22708b25daf7f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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3 . 已知数集
具有性质
:对任意
,
与
两数中至少有一个属于
.
(1)分别判断数集
与
是否具有性质
;
(2)求证:
;
(3)给定正整数
,求证:
,
,
,
组成等差数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e7cdd844688e7a1b08f8ed3792760a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29bcdd8c05cd04f46c6f4ba8aa3cb1d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c981868188750ab216366b8272d4b35f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3541598c0e0e6d5050c5a562515c430e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b13ee542834ccbb57fcc55b1680ca9db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4489d9b83072184c0e1d6b09be50ca.png)
(1)分别判断数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edd14cb27cbdb43c432f7493c34575c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ec8506f11fb704c94772de34e05381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29bcdd8c05cd04f46c6f4ba8aa3cb1d0.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86e6fc2c0740f9ff797037bcd1409768.png)
(3)给定正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fee7283a714563ad255f3ef9ac1a30.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/07096af3b99fd1cb11c31f19a2c6408e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
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2023-12-20更新
|
408次组卷
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4卷引用:北京市海淀区中央民族大学附中2024届高三上学期12月月考数学试题
北京市海淀区中央民族大学附中2024届高三上学期12月月考数学试题(已下线)专题1 集合新定义题(九省联考第19题模式)练(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北京师范大学燕化附属中学2023-2024学年高二下学期期中考试数学试卷
4 . 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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5 . 已知正整数
,对集合
及其每一个非空子集
,记
,其中
,定义一个运算“交替和”
.例如:对于集合
,
.则当
时,集合
的所有子集的“交替和”的总和为_________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b315521745464732c0eb8c9bcc25aea3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65e3ec8c5a00825669e691119089547c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f3761763cd8c778f93f11f2a89cde6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a8cef566b38e8fff93f7af90b2909fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83dceca5fe4f16a0455490f0023bd9a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a4b06447ce9c66708dd74512960f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764eff906937f9b1fb58e5abfb2eb8a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b315521745464732c0eb8c9bcc25aea3.png)
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6 . 对于平面上点
和曲线
,任取
上一点
,若线段
的长度存在最小值,则称该值为点
到曲线
的距离,记作
.下列结论中正确的个数为( )
①若曲线
是一个点,则点集
所表示的图形的面积为
;
②若曲线
是一个半径为
的圆,则点集
所表示的图形的面积为
;
③若曲线
是一个长度为
的线段,则点集
所表示的图形的面积为
;
④若曲线
是边长为
的等边三角形,则点集
所表示的图形的面积为
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72a421e5d3b3752b9e1f04c30799bace.png)
①若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26bee18b0b9fdb657956eaeb43023288.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20b3a91ccf6028608cd03df7072f6536.png)
②若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d079a36153052a0fbdf35b2a2bd3b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e95a79288fcb7e47fba4410722e2bc6.png)
③若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d079a36153052a0fbdf35b2a2bd3b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7b0c7fb6244310e468d5092cf0dba4f.png)
④若曲线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d02ea8c4988c5c28ab93f0d70fb55a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d079a36153052a0fbdf35b2a2bd3b67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebceb91a9f116f8f5308bcad3020bae4.png)
A.1 | B.2 | C.3 | D.4 |
您最近一年使用:0次
2023-11-23更新
|
436次组卷
|
6卷引用:考点3 与集合相关的新定义问题 --2024届高考数学考点总动员【练】
(已下线)考点3 与集合相关的新定义问题 --2024届高考数学考点总动员【练】(已下线)专题1 直线与圆的位置关系【练】(压轴小题大全)北京市陈经纶中学2023-2024学年高二上学期期中考试数学试卷浙江省湖州市第二中学2024届高三上学期期中数学试题(已下线)专题02 直线和圆的方程(5)(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)
23-24高三上·北京·期中
名校
7 . 已知集合
,集合
,且满足,
,
与
恰有一个成立.对于
定义
,以及
,其中
.
例如
.
(1)若
,
,求
的值及
的最大值;
(2)从
中任意删去两个数,记剩下的数的和为
,求
的最小值(用
表示);
(3)对于满足
的每一个集合
,集合
中是否都存在三个不同的元素
,
,
,使得
恒成立?请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d7759d382dfd33b5a08fa4592b5178.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0b8b719c20b1888fe2cb3d454419ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b18a24af2916239dfd649c37cf27770.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9501ed9d84c443aee21eb55f5ce385f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1322ce3a3343914627e90d26e39e40f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0179662fc6cf8112ec0c42c8f6cdf53a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6288cc5be15de868fe6c1f6fef6a309.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12b0e3b00fe47801afb53ec56706c21a.png)
例如
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63af360678d1ba9333fcb05dc46b08b9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fac3649308b528fd56545ba102dc42d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0ea63a47001e23b4d244777be6d2719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee3ab6dd619d6d614608a2087dff6186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c11ec2069caa7f0d750971ff67fe1288.png)
(2)从
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/726c816d07c8c3adc7fa610e3d0335e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)对于满足
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5f5178625a53061308733d7f0393199.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/168b3e4b1d6f04226fa2687a72a268b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca4ff0af96ea467337cb30c4c765b5f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276509f01529d982ab21e479a4619268.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbd1d5178f23e7be5a3da0399a770404.png)
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23-24高三上·上海浦东新·期中
名校
8 .
是正整数集的子集,满足:
,并有如下性质:若
、
,则
,其中
表示不超过实数
的最大整数,则
的非空子集个数为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793b9b22f06177785864b283ff38d61e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dab92728b35ed5798e07a2b0095bfcc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b958eb9d7d286a1c6daf4711f304b9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c4f5908d6a1217e493ed7586b6964dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
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解题方法
9 . 对非空整数集合M及
,定义
,对于非空整数集合A,B,定义
.
(1)设
,请直接写出集合
;
(2)设
,
,求出非空整数集合B的元素个数的最小值;
(3)对三个非空整数集合A,B,C,若
且
,求
所有可能取值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ef96396caccbf2f959e9d233f060317.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff97e9f30e667709fc602b8dcc7523e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ef9a7e4979cb5c0e8f4ff92af351dc.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/679a7e0835c8200dc4975baf3e89d058.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/048b9f7e0aa69227773cb9d9b7bb6244.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2463f83ca63b14937787c99fcd1958d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e79b6edfd5f5bb3866599fcb2aa6f3.png)
(3)对三个非空整数集合A,B,C,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22fcff4bcf7a1e6652774a5aee4519c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72944e6acd452acaa213f6ae23ae9a2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5a6e825795c828f8281e93607d0e1b1.png)
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|
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4卷引用:2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题变式题16-19
(已下线)2024年1月普通高等学校招生全国统一考试适应性测试(九省联考)数学试题变式题16-19(已下线)专题1 集合新定义题(九省联考第19题模式)讲北京市清华大学附属中学2023-2024学年高一上学期期中考试数学试题江苏省徐州市沛县第二中学2024届高三下学期期初测试数学试题
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解题方法
10 . 已知
元正整数集合
满足:
,且对任意
,都有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/453e81d3dacb5e2e527b8daebcf19fc9.png)
(1)若
,写出所有满足条件的集合
;
(2)若
恰有
个正约数,求证:
;
(3)求证:对任意的
,都有
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5fcfc55b56f04ac530871067e5ce71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b5ba5a3d48da1d7f3952c013a5ee0de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/139ae1c18e2cded6d04808a3aa24c6bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26c7cb34e9b5c1ecd75ff1966e882ae7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/453e81d3dacb5e2e527b8daebcf19fc9.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/039e4fe671d61e59b96ee525c9df43e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7334c46af837676ada9575630a48d60f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7979eac1b85b681ba94bf8d833c5c6e7.png)
(3)求证:对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fa673986cef1873a421a572abf9ed71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57cee7a77f2f4592cb5c96245ff025f8.png)
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