1 . 设
,而
为S的一个8元子集.求证:
(1)存在非零自然数k,使得方程
至少有3组不同的解;
(2)对于S的7元子集
,(1)中的结论不再总是成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06018a972e02a43b95f5c78aca784610.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7bf032b085c415f6ce188bd9be0afe6.png)
(1)存在非零自然数k,使得方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aae335e59364467c2ad9d5602f220af2.png)
(2)对于S的7元子集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7c1f4838725717094b55f5d82a3e2ca.png)
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2 . 设函数
,集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c563b30e0b214a3f2996e6c4c1d5ccf.png)
(1)证明:
.
(2)当
时,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82978e25c21f5153ffe233a2720880c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c563b30e0b214a3f2996e6c4c1d5ccf.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ad78dc8b8aed907b4fe9640c997454.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37d648d3fa1544e9de76c9d0ff3b021a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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2023-08-16更新
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3卷引用:黑龙江省哈尔滨市第九中学校2024届高三上学期开学考试数学试题
黑龙江省哈尔滨市第九中学校2024届高三上学期开学考试数学试题黑龙江省哈尔滨市第九中学校2023-2024学年高三上学期开学考试数学试题(已下线)第一章 集合与常用逻辑用语(单元测试)(能力卷)--高一数学同步精品课堂(人教B版2019必修第一册)
3 . 已知
为一个数集,集合
.
(1)设
,求集合
的元素个数![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(2)设
,证明:若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0beedd42faff3d322366bd04438cb1f5.png)
(3)设
,
,
,且
,
,若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2638a68134a07d20bfb878a7f48e9784.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa8d5b5b71da38bf620a5db84ebe41c.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/993161b4e3fc33b8ff76a9372451c597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32ca6fa9955690cec01db601e3abce0c.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3aae73f5f1742524665a8c83328b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0beedd42faff3d322366bd04438cb1f5.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860ebb6f76cd3cb9a265dfc233002a13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a83c3f3b7aac5fe26a6f528ba8165f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7082c8fbddfb0a20f88332c32d2cd649.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc813e97b6351601ed106acfb91160d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79e026ad6c26ffc10e03d7d33d5f204a.png)
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解题方法
4 . (1)已知
,比较并证明
与
的大小.
(2)求方程
的解集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3395a33ac00b409696a33a8a62f76c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29ebc856291255f2d4a6c20b982a2442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00c3cdabd2cc1450d8a6aa5bf4481f53.png)
(2)求方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51a39b01784d7fc768ed683aebd9e46c.png)
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5 . 对正整数
,记
,
.
(1)用列举法表示集合
;
(2)求集合
中元素的个数;
(3)若
的子集
中任意两个元素的和不是整数的平方,则称
为“稀疏集”,证明:存在
使得
能分成两个不相交的稀疏集的并集,且
的最大值为14.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a1b26aa2a8eae39c45ab0b5e4b0888.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d38f7005b0c6ade2daf8bf39c17c957.png)
(1)用列举法表示集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
(2)求集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34e24e843f778f53af4f3c9e25faa809.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf83e20035c3afd6d26ebfd53d768a70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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2022-10-13更新
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4卷引用:上海市华东师范大学第三附属中学2022-2023学年高一上学期第一次阶段检测(10月)数学试题
上海市华东师范大学第三附属中学2022-2023学年高一上学期第一次阶段检测(10月)数学试题上海市浦东新区新川中学2022-2023学年高一上学期期中数学试题(已下线)第一章 集合与逻辑(单元基础卷)-【满分全攻略】(沪教版2020必修第一册)(已下线)专题01集合及其表示方法2-【倍速学习法】(沪教版2020必修第一册)
名校
6 . 设集合
,
.
(1)若
,求集合
和
(用列举法表示);
(2)求证:
;
(3)若
,且
,求实数
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ddf853aaab07a537a37cd80bca0c4dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/237752297babca42d3c3709b4cbfb34d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267e6d77aabbebe52e7aca993368d874.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2333f966f6ec29f0661f93d99b055cd5.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f31fff6c9a52fd12a5ea3408b2bc41c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad96b50521f4bbf3d436d05dc258083d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
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7 . 设集合
.
(1)将集合
中的元素进行从小到大的排列,求最小的六个元素组成的子集
;
(2)对任意的
,判定
和
是否是集合
中的元素?并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffed7c978d85a42aef8161f721d5bdf7.png)
(1)将集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)对任意的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de590ef958bd3d167242c795007b8b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/450398974b1561ca801e102e16df6789.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28b6ab702f8e93cc1e680a7d7af06786.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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2021-10-10更新
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6卷引用:上海市延安中学2021-2022学年高一上学期10月月考数学试题
上海市延安中学2021-2022学年高一上学期10月月考数学试题(已下线)专题01 集合与常用逻辑用语常考基础题型-2021-2022学年高一《新题速递·数学》(人教A版2019)(已下线)1.1集合的表示方法(第2课时)(已下线)第01讲 集合的含义与表示(4大考点12种解题方法)(3)上海市格致中学2023-2024学年高一上学期10月月考数学试题第一章 集合与逻辑(知识归纳+题型突破)-速记·巧练(沪教版2020必修第一册)