1 . 由无理数引发的数学危机一直延续到19世纪,直到1872年,德国数学家戴德金从连续性的要求出发,用有理数的“分割”来定义无理数(史称戴德金分割),并把实数理论建立在严格的科学基础上,才结束了无理数被认为“无理”的时代,也结束了持续2000多年的数学史上的第一次大危机.所谓戴德金分割,是指将有理数集
划分为两个非空的子集M与N,且满足
,
,M中的每一个元素小于
中的每一个元素,则称
为戴德金分割.试判断下列选项中,可能成立的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/316ecb1589c3cc179e2f62507020771e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/252b52fe186ca8f10398dcd32e9ce394.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4815b1d16a7ae485ff0bba0b397e893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb4a195a4245b05754edb54660eccc9b.png)
A.![]() ![]() |
B.M没有最大元素,N有一个最小元素 |
C.M有一个最大元素,N有一个最小元素 |
D.M没有最大元素,N也没有最小元素 |
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解题方法
2 . 定义:有限集合
,
则称
为集合
的“元素和”,记为
.若集合
,集合
的所有非空子集分别为
,
,…,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e46f01c4a5c57ded953c5796f318dd.png)
________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6e701c68e1038582c4ef1eceb87115.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6559598727fb120a5cdbf4f15510615d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03b5214a796412b3df9f716da0bf339b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/685fdef2e4872004134d6bb7d1cf8f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2708fa6298e52f617383efc175b71ddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf99487d7860d017c0747ff966edfd77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29e46f01c4a5c57ded953c5796f318dd.png)
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2024-03-07更新
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265次组卷
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3卷引用:贵州省贵阳市清华中学2023-2024学年高二下学期4月月考数学试题
名校
解题方法
3 . 已知全集为R,对于给定数集A,定义函数
为集合A的特征函数,若函数
是数集A的特征函数,函数
是数集B的特征函数,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ef299b559eb12ac8b45b8dc056fb69a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4669810732b633b60dbeaf0bf57204f6.png)
A.![]() ![]() |
B.![]() ![]() |
C.![]() ![]() |
D.![]() ![]() |
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2024-02-23更新
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314次组卷
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2卷引用:湖南省株洲市第二中学2021-2022学年高一上学期第三次月考数学试卷
名校
解题方法
4 . 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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5 . 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
|
400次组卷
|
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
名校
解题方法
6 . 设整数集合
,其中
,且对于任意
,若
,则
.
(1)请写出一个满足条件的集合A;
(2)证明:任意
,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eab8bf709a13b3d6cea5bf2b05c92019.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d1c5d7e92a7d6c61e007cd9313b1b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b836bde5106e78caeb728ff3353bee7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/269c7a915fc171ac7ad84c09883a6dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1250b7f54f3a23a5e52b2e4aa0fc0050.png)
(1)请写出一个满足条件的集合A;
(2)证明:任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/361b9733dc8c4896ce0501d1a3ddf3a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7848c89302a41e9576530313fc3e61b8.png)
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7 . 对于正整数
,定义
.对于任意的
,称
为
的第
个分量,称
是
的一个“协同子集”.如果
同时满足:①
的元素个数不少于
;②对于任何
、
、
,存在
,使得
、
、
的第
个分量都是
.
(1)对于
,若
是
的一个恰好含有四个元素的“协同子集”,且其中两个元素是
和
,直接写出另外两个元素;
(2)证明:若
是
的一个“协同子集”,则
的元素个数不超过
;
(3)证明:若
是
的一个“协同子集”,且
的元素个数恰好是
,则存在唯一的
,使得
中所有元素的第
个分量都是
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d3dc6cad699aa2713482c9f4306802.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e6aa5d5b774230400311326853ed898.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97ea8f47d8d8d9e1832d52b1c7425450.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c05b9832b09731a574d4a4adf7448de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](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/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a86c79fb771a07a413c755e4295b160.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01433c50807a7878f60c05f43c3fa652.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a829fdd8ec0f3b7ede883cf2c3e53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0d8b9ad2fcfff3dd546c5fdbedfe6238.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7600d2cfbdc6146db96cc545706004f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
(1)对于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9154d05e636d76f2726e226a5ef3d7fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98b01af54049b27ca6e8159518b7b18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/968b3f41b9a2f481de4b9d95547c5423.png)
(2)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f1ad18371ec533aeac27cf1fad95c1.png)
(3)证明:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e00d8f90655e6341907aa9c7c62d4398.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f1ad18371ec533aeac27cf1fad95c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96e1f6f6f70deeead9aa004fe0697323.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
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8 . 用
表示非空集合
中元素的个数,定义
,已知集合
,则下面正确结论正确的是( ).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3b13b79a46dfef4718ebde60a90972.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68cbf4edfaa2c40675feb1481b6ade25.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3ff69c3f01921cd51df0064a871842a.png)
A.![]() |
B.![]() |
C.“![]() ![]() |
D.若![]() ![]() |
您最近一年使用:0次
2023-11-27更新
|
219次组卷
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2卷引用:山东省德州市万隆中英文高级中学2023-2024学年高二下学期6月月考数学试题
解题方法
9 . 设集合
,
,
,
,
,
中至少有
个元素,且
,
满足:①对于任意的
,
,若
,则
;②对于任意的
,
,若
,则
.下列命题不正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53a9e674fa61f737f656d11b3c88dca5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a5a2f21009e01e4fb578625202e9a17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45c8b2714e2f6ddfdd6b05d3b4de1149.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/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d81972b1768d827ba3083f96a273412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a5abe56c019ac914e1fcde1865a747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5718e9c8baa106b447f9fae23e730de.png)
A.若![]() ![]() ![]() ![]() |
B.若![]() ![]() ![]() ![]() |
C.若![]() ![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() |
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10 . 已知集合
(
),
表示集合
中的元素个数,当集合
的子集
满足
时,称
为集合
的二元子集,若对集合
的任意
个不同的二元子集
,
,…,
,均存在对应的集合
满足:①
;②
;③
(
),则称集合
具有性质
.
(1)当
时,若集合
具有性质
,请直接写出集合
的所有二元子集以及
的一个取值;
(2)当
,
时,判断集合
是否具有性质
?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e75a2aec5a4cd169b811cef69c9508a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e819b87f90651d89fcd258c276294e43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a50d86b46e0771672997f4ec429e91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e819b87f90651d89fcd258c276294e43.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/294f5ba74cdf695fc9a8a8e52f421328.png)
![](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/7a0d7559d8dfa8236ca9d4b1853fbdec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf22d7d1a965bda25168a233fb6290c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a01cb028fb377008e680de7231c72e01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e1f0121616edbbbab89eed23edd9109.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65bad3343f0e29460ead5b54771f362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67716ac738ee2911a69bf4063110a5bd.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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