解题方法
1 . 已知集合
是由某些正整数组成的集合,且满足:若
,则当且仅当
其中
且
,或
其中
且
.现有如下两个命题: ①
;②集合
.则下列选项中正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09256752badab8d69ae679796896ed97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d69ad2e6fb245dd796a90069cdecf5bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4e2c7d495c75424714210323455d9ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb4e301bb40ce2f30a10cd44ddb5a95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f44754f1a3fb7f49c2f1bc8d55a78d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51e6e9364f9bff37c962556537a0a7a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0908dc92cc1b6c38fa8114c0e122c4eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d00c07c7865e822c502ffb72ae0406d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07b29af0b42059c645d2164532cd286c.png)
A.①是真命题, ②是真命题; | B.①是真命题, ②是假命题 |
C.①是假命题, ②是真命题; | D.①是假命题, ②是假命题. |
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2 . 若项数为
的数列
满足:
,且存在
,使得
,则称数列
具有性质P.
(1)①若
,写出所有具有性质P的数列
;
②若
,写出一个具有性质P的数列
;
(2)若
,数列
具有性质P,求
的最大项的最小值;
(3)已知数列
均具有性质P,且对任意
,当
时,都有
.记集合
,
,求
中元素个数的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2adcce353c3ced76645069760bdf8fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad107fc0e55e4b35b2b25b10f75f4e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b96b51d9b57fb8f9577f575261c08206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0265ce589cdf60588c9f86671a256992.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc147c3c0863d22533a820cc469bfb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0818b44478f6d1d972aa5bf6dd4d3a0c.png)
(1)①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a215612787e43d28bfebc840c3903b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04b56e44e4f0424a2b7a45567120a2e4.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c1ddc5b7f288c044f394fd7055e2564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e47cd514b2920609e3781c87df6ab70.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e7aa41f0d2bd18825f741a3cc2adef0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e645ae0b78ad4ca300e3889ca3f9bcce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e645ae0b78ad4ca300e3889ca3f9bcce.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/719c07413b9a867b83f27f5eed752846.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6119a01b55a29adabf43ddb2a5211e8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0504794eebd6814a3138e796666bca01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/191929f42f9053e7355a1a9068ba072f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de5975dbb4789b4b2d8f2f0eb3abb4da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdc47ee887b63cdc46fdb17dcb75f2f4.png)
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3 . 对任意数集
,满足表达式为
且值域为
的函数个数为
.记所有可能的
的值组成集合
,则集合
中元素之和为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d020cd453031ae9eede7961ec78f21a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4702a6ec0cbff878d6dc9cbd6596433.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
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4 . 正整数集合
,且
,
,
中所有元素和为
,集合
.
(1)若
,请直接写出集合
;
(2)若集合
中有且只有两个元素,求证“
为等差数列”的充分必要条件是“集合
中有
个元素”;
(3)若
,求
的最小值,以及当
取最小值时,
最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/59003c6febf1a45fc85bc475a3a17b8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c002c44d45907aad22da19859193270b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75fd9ec9c065d4337a8b1ebf2abc6a1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dd8ff87ff3fef45414f594ae017351e.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98bef20fc87980ef78c7f8eebd15f308.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
(2)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d02a8555da4dbbc7820a50a95b071ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/629455b181f187fe9b0edd9d66431ef0.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/143d9d0954abea7adb3412216fe8daf7.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/96abfe2da27a63e6affb19a0c80236d9.png)
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5 . 从集合
的非空子集中随机取出两个不同的集合A,
,则在
的条件下,
恰有
个元素的概率为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff1f111415ece207aad8e2320185330d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7dcbd7f44645b8c707627ab0795e18b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fdbfa7a63fdf5717d40c8c9a73ec160.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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6 . 已知X为包含v个元素的集合(
,
).设A为由X的一些三元子集(含有三个元素的子集)组成的集合,使得X中的任意两个不同的元素,都恰好同时包含在唯一的一个三元子集中,则称
组成一个v阶的Steiner三元系.若
为一个7阶的Steiner三元系,则集合A中元素的个数为_____________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe0953f915a38732687056d9e2dee04c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e6ce9503fa100664e4aa127f513a167.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8721733e3975550483a89a78203dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8721733e3975550483a89a78203dd.png)
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10卷引用:湖北省2023届高三下学期四月调研考试数学试题
湖北省2023届高三下学期四月调研考试数学试题专题01集合与常用逻辑用语(已下线)高一上学期第一次月考填空题压轴题50题专练-举一反三系列(已下线)高一上学期期末考试填空题压轴题50题专练-举一反三系列江苏省南通市新高考2024届高三适应性测试数学模拟试题河南省信阳市新县高级中学2024届高三考前第三次适应性考试数学试题(已下线)第01讲 集合(七大题型)(讲义)(已下线)集合及其运算(已下线)新题型01 新高考新结构二十一大考点汇总-1湖北省武汉市第十一中学2023-2024学年高二下学期6月考数学试题
7 . 已知
为正整数,集合
具有性质
:“对于集合
中的任意元素
,
,且
,其中
”. 集合
中的元素个数记为
.
(1)当
时,求
;
(2)当
时,求
的所有可能的取值;
(3)给定正整数
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0453075033462de2767a6a0ea21fd10b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bcfa120f92eebe8b4d2a98b5e4ee5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0114bfa275e57023f2ebb88274fe8fba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2941376f87609dc0c57675c1da9d3e32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de23b532e3b47ff7bc43ede3c11ae94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9bd23c1935ae3b80cca9151c23ae92a.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2d3df37e73a8abea815f37dbb3fff5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9bd23c1935ae3b80cca9151c23ae92a.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294250dbd576bff3da0a1456cb9a88a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b622e69397f1fa6b3a8e4e3f947642.png)
(3)给定正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9bd23c1935ae3b80cca9151c23ae92a.png)
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7卷引用:北京市延庆区2023届高三一模数学试题
北京市延庆区2023届高三一模数学试题专题12压轴题汇总(10、15、21题)专题11计数原理与概率与统计专题01集合与常用逻辑(已下线)北京市丰台区2023届高三下学期3月一模数学试题变式题16-21北京卷专题26计数原理与概率与统计(解答题)北京卷专题02集合(解答题)
8 . 已知集合
.若对于集合M的任意k元子集A,A中必有4个元素的和为
,则称这样的正整数k为“好数”,所有“好数”的最小值记作
.
(1)当
,即集合
.
(i)写出M的一个子集B,且B中存在4个元素的和为
;
(ii)写出M的一个5元子集C,使得C中任意4个元素的和大于
;
(2)证明:
;
(3)证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40fcb12b12b124d437768cfc9bc3a23d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e48acb84d3c7fb67ed275add00ea178.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be604061cf1591f7069472269d4c9719.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72790d6daf2b07dfbbbc7a340eeb6ffb.png)
(i)写出M的一个子集B,且B中存在4个元素的和为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
(ii)写出M的一个5元子集C,使得C中任意4个元素的和大于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acbc6a613224461ade69362d46550474.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2c5470cb4bb49c443a9568d79eadd39.png)
(3)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/831fa5ef5f6cea7664a4d385b7c98105.png)
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北京市门头沟区2023届高三综合练习(一)数学试题专题12压轴题汇总(10、15、21题)专题01集合与常用逻辑(已下线)北京市丰台区2023届高三下学期3月一模数学试题变式题16-21北京卷专题02集合(解答题)(已下线)专题22 新高考新题型第19题新定义压轴解答题归纳(9大题型)(练习)
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解题方法
9 . 已知数列![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f155247fc662b89c545ea34094b778d6.png)
,
,…,
的各项均为正整数.设集合,
记
的元素个数为
.
(1)若数列
1,1,3,2,求集合
,并写出
的值;
(2)若
是递增数列,求证:“
”的充要条件是“
为等差数列”;
(3)若
,数列
由1,2,3,…,11,22这12个数组成,且这12个数在数列
中每个至少出现一次,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f155247fc662b89c545ea34094b778d6.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/8d8e6e9e456eb7ed9de302990075d1c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3093d1159a994b153db26d1ba23c567f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a318eb5a4da016d3d993175e845a90ab.png)
(1)若数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f155247fc662b89c545ea34094b778d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a318eb5a4da016d3d993175e845a90ab.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be63af01fc637c108801b34882acc1a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/807511534842b45a837c0fe12754345e.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/a318eb5a4da016d3d993175e845a90ab.png)
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10 . 给定正整数
,设集合
.对于集合
中的任意元素
和
,记
.设
,且集合
,对于
中任意元素
,若
则称
具有性质
.
(1)判断集合
是否具有性质
?说明理由;
(2)判断是否存在具有性质
的集合
,并加以证明;
(3)若集合
具有性质
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a839578f0b23c8aeba01730563a643e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da8a33568ded09f3450bb153b0e5697.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714ab9e5a6949c90c9bfdd118cfabeb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35e477c52dfbfb80f1fc315143c8b06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1368a045ba80f97383f3d9d7fcdc8f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cd94234d029d89c7b788b6d1e225db6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9855cb665c7f3785a17718be10538af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a2f08194bb663f1a086fa2f555ebf43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faa6d3ee76dcca88508ec0921f1adf0f.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbd6650ab1ac1f7426ec68c729671c41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c581b105a7e14eae97d650ae73adf710.png)
(2)判断是否存在具有性质
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a742b9bb0812b7bb895851cc5a06fa1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faa6d3ee76dcca88508ec0921f1adf0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b74258d0a66cadc32ba68697abca894.png)
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2023-03-27更新
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1986次组卷
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13卷引用:北京市西城区2023届高三一模数学试题
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