名校
解题方法
1 . 若正整数集
的非空子集
满足:至少含有2个元素,且任意两个元素之差的绝对值大于1,则称
为数集
的超子集.对于集合
,记
的超子集的个数为
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26974cd38196094faf713e10668d42a2.png)
______ ,
与
的关系为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12934b832f1484fb88b13974e50cad4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12934b832f1484fb88b13974e50cad4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10937bdfe1d64b5f53b598b377f94312.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26974cd38196094faf713e10668d42a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a69725055853cf456a3c92632bd54dbe.png)
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名校
解题方法
2 . 已知全集
,集合
,
,则图中阴影部分表示的集合的子集个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c997afeb2e9be9c16822ab852dfafe6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5990fe8270a3cd33f089e889f68792db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c5df737e4d0cc03539294d74eef9c35.png)
A.2 | B.4 | C.8 | D.16 |
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3 . 若集合
的非空子集
满足:对任意给定的
,若
,有
,则称子集
是
的“好子集”.记
为
的好子集的个数.例如:
的7个非空子集中只有
不是好子集,即
.记
表示集合
的元素个数.
(1)求
的值;
(2)若
是
的好子集,且
.证明:
中元素可以排成一个等差数列;
(3)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f5c44de003475d3466981293cf5e47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/887982e3735dd7ca13293338a12df593.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c469f6345826410959ea09d7e3192e20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0586ec8d1d9796fb80a1250e2c0a4b0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b26bbb11e932ddb26a9088e7fc33e87b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27c562f247c1d691158f4038a030574c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68eafd45c1ec4b414d3553dabd8c2848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37765d2927d24d4b582423c843aebcd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b32a859898e9905e0524d3a982eb34b6.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ada01c2a8b4d92df94834a6a3929673.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(3)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79df6a6d73a058d13632a726c2308d66.png)
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解题方法
4 . 已知集合
,
,
,若
,
,
或
,则称集合A具有“包容”性.
(1)判断集合
和集合
是否具有“包容”性;
(2)若集合
具有“包容”性,求
的值;
(3)若集合C具有“包容”性,且集合C的子集有64个,
,试确定集合C.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe38d1b1e18720a878bd7442c8f094de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e145b6046bc80d0ffecc61ac67c87ca1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e63c91626ffa91e590925e6f206c3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/957d41dbe52b49c3a7339e3519a3fe84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c46de01c5104b9112a688df37eadb000.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bac46c6c52fc8b8e8f76084352e1893f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b3ed03b0f8fb8b88d7edf6165345c6f.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6acf45d9d64697db902fe8faa15394d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c925be255ca736a53b24d13ddede1a86.png)
(3)若集合C具有“包容”性,且集合C的子集有64个,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f65f44d4f1341a7d80897a54a6778fae.png)
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名校
5 . 若集合
,集合
,其中
,则称集合
是集合
的一个“
元子集”.若“
元子集”
中的元素
满足对任意
,恒有
,则称
为
的一个“个性独立子集”.已知集合
,集合
是
的一个“个性独立子集”.
(1)求所有满足条件的集合
的个数;
(2)若
且互不相等,证明:
为定值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b7c647eb8e6be44689333e1a6f1f8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/473ed4e9293f5066fe367cfa541afa11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5d87fa699e2aca137f69e4ac0883dbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9304e71a623c4412188a800046a970d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a4aecf0b7edfa826f73389322a52cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86f2c01531dd639f0ad3da0098febbb3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/effaffc8a4d01f865a79933d3c337060.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)求所有满足条件的集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ab3fa353ea21d2dea55e32a354e3dfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f0736f73b81bec1c5d4efe968cfbe2a.png)
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名校
解题方法
6 . 从集合
的子集中选出
个不同的子集
,且
,则选法有_________ 种.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ad8810256459629f91882918fd45597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ae1567d8f98fabc1a3948f8602cc5e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ad78dc8b8aed907b4fe9640c997454.png)
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名校
7 . 斐波那契数列(Fibonacci sequence),又称黄金分割数列,因数学家莱昂纳多·斐波那契(Leonardo Fibonacci)以兔子繁殖为例子而引入,故又称为“兔子数列”,指的是这样一个数列:1、1、2、3、5、8、13、21、34、…,在数学上,斐波那契数列以如下递推的方式定义:
,
,
(
,
),已知
,则集合A中的元素个数可表示为
,又有
且
.
(1)求集合A中奇数元素的个数,不需说明理由;并求出集合B中所有元素之积为奇数的概率;
(2)求集合B中所有元素之和为奇数的概率.
(3)取其中的6个数1,2,3,5,13,21,任意排列,若任意相邻三数之和都不能被3整除,求这样的排列的个数.(如排列1,2,3,5,13,21中,相邻三数如“1,2,3”(“3,5,13”、“5,13,21”),和能被3整除,则此排列不合题意)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b065334d8f60c49f4bd3d9f1373fe4cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f966272f7781790ff27e40db6b525253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6a404164c8d199f60d183a59b3647cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb976cc41026ce1540505e9c5f9e81a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82e5ee1d004ae893eb0190b6e9a4c6c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/caf22d7d1a965bda25168a233fb6290c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3331942d1f39489803a81d76844cc442.png)
(1)求集合A中奇数元素的个数,不需说明理由;并求出集合B中所有元素之积为奇数的概率;
(2)求集合B中所有元素之和为奇数的概率.
(3)取其中的6个数1,2,3,5,13,21,任意排列,若任意相邻三数之和都不能被3整除,求这样的排列的个数.(如排列1,2,3,5,13,21中,相邻三数如“1,2,3”(“3,5,13”、“5,13,21”),和能被3整除,则此排列不合题意)
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2024·全国·模拟预测
名校
8 . 已知集合
,
,则
的子集的个数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/118486ca62d7e1b53302dd8cfe075d12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00c2639128678d39f539043e781cd63f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fdbfa7a63fdf5717d40c8c9a73ec160.png)
A.3 | B.4 | C.8 | D.16 |
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名校
9 . 已知,集合
,若集合A恰有8个子集,则n的可能值的集合为
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2024-03-14更新
|
594次组卷
|
2卷引用:上海市上海师范大学附属中学2023-2024学年高三下学期3月月考数学试卷
名校
10 . 设A是正整数集的一个非空子集,如果对于任意
,都有
或
,则称A为自邻集.记集合![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c9e547b17582b99e548037172eeff3.png)
的所有子集中的自邻集的个数为
.
(1)直接写出
的所有自邻集;
(2)若n为偶数且
,求证:
的所有含5个元素的子集中,自邻集的个数是偶数;
(3)若
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ed006b944ea64f970fee46e2f558467.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/417abc71b8bee465746db0a35e776f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ca2371b88985463ba25e4ec1ea453d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c9e547b17582b99e548037172eeff3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/677fd74842cbce34aed7073cebbd9c58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
(1)直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e47cd514b2920609e3781c87df6ab70.png)
(2)若n为偶数且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831608f09609c37f757f5bfcd01253f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3cfeacc29e6a61c5b3b4e439c0a91df.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04aba3402e1d191ff96adda7c4af70ef.png)
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2024-03-12更新
|
448次组卷
|
2卷引用:北京市第八中学2023-2024学年高三下学期3月月考数学试题