1 . 集合论在离散数学中有着非常重要的地位.对于非空集合
和
,定义和集
,用符号
表示和集
内的元素个数.
(1)已知集合
,
,
,若
,求
的值;
(2)记集合
,
,
,
为
中所有元素之和,
,求证:
;
(3)若
与
都是由
个整数构成的集合,且
,证明:若按一定顺序排列,集合
与
中的元素是两个公差相等的等差数列.
![](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/88c7a43079a55f6a53b1307b2b04b55e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93cf60f60fedb84bb62a0c00276908ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ce69cd33d105ce280170f0cd0513026.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed0081de4e04574dd0884c4e6077fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d651573ff643d295dcceafdb6f1249d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f42500dfa5011086d43ef7e6dac58271.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b27c12cad9040ae9698895e43903747.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(2)记集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95c9e547b17582b99e548037172eeff3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41f7f9dc32fa86d097de2b7d78b6b487.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/594e60168219fdebb98b45493de0128a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/96abfe2da27a63e6affb19a0c80236d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcc31dcdb99754fc452ff2b92a2fb8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f093c61867ee4ce75f951d46b9b123.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dc9ec58912d76aabf278faa7bf06e45.png)
(3)若
![](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/ef96e432405a1037b5aea7514715e52d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa740177330d445b0d506f3b53f9ad2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
您最近一年使用:0次
2 . 设
,
是非空集合,定义二元有序对集合
为
和
的笛卡尔积.若
,则称
是
到
的一个关系.当
时,则称
与
是
相关的,记作
.已知非空集合
上的关系
是
的一个子集,若满足
,有
,则称
是自反的:若
,有
,则
,则称
是对称的;若
,有
,
,则
,则称
是传递的.且同时满足以上三种关系时,则称
是集合
中的一个等价关系,记作~.
(1)设
,
,
,
,求集合
与
;
(2)设
是非空有限集合
中的一个等价关系,记
中的子集
为
的
等价类,求证:存在有限个元素
,使得
,且对任意
,
;
(3)已知数列
是公差为1的等差数列,其中
,
,数列
满足
,其中
,前
项和为
.若给出
上的两个关系
和
,请求出关系
,判断
是否为
上的等价关系.如果不是,请说明你的理由;如果是,请证明你的结论并请写出
中所有等价类作为元素构成的商集合
.
![](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/28b93f7aa7ba32c9dad112ae7caa10d2.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/76b076845d2b97a8b09807f232000aa0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.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/558b4d40179245aa327521eeff8c2574.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a15d37048f967e9420c3d117d8231d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31a7c9c05b4d3eac6461747017dcb8cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7902d1a9d757df4d9bc35d45e16d892.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba85c8b02a51af9a7f2121f6888de7df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b548de80bcd12b1bc37081ac69a7431b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a15d37048f967e9420c3d117d8231d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a825fd8b77fbb7342cd408968fb70ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/46ea1419908c307c68726c8266022584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a15d37048f967e9420c3d117d8231d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/050a5bbe5ed5a5ffb338f6754a884fc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b04042e0bf9c6985ffc72e63134b6416.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d65c189a79078617afd2f9a455ccea4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5035c62eda0e9238d517fea6b5bb6f0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ce240043bb6d7e24a09954f7c72a14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62d4afc4786dd071158544fcd1f5b132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1169b97c3532be1b2a67f053a7d2c807.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34bc98fb66e6c435ee3f3ae838b56666.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](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/e295975b6e7d533fca11356ef38f0877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/994598ce57f0289a3cb374740e431235.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf81dd43d0ab4be39344ef96aa2b25e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b71af6590f0f369c164a054a8b63bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27db6e128a3c29b8df7f8743546bb8db.png)
(3)已知数列
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b36e3ca48d6825b91d99dc49861584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29b55a10b9c9abf002dc82b2951251b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c1a134d2f29b023f3355aa5b4af457d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76aef4cdcb5af742ce28003b7b6c8c20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/451eedd2b6db5a8233816f51788f54a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc9efeb4455e30293d412938eeea85d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8ad9141b70ad7eadb9dabec40186f40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b602d8facb00e929bd7b7dbe607d724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc868066533c40faab358a931a6aeb84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31be75a542de7085c49dddc2403de62e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc91509afee726c4279a7767da66dadb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b602d8facb00e929bd7b7dbe607d724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b602d8facb00e929bd7b7dbe607d724.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f2368d861c70f08c2721e8181954cd.png)
您最近一年使用:0次
3 . 对于数集
,其中
,
,定义向量集
,若对任意
,存在
,使得
,则称X具有性质P.
(1)设
,请写出向量集Y并判断X是否具有性质P(不需要证明).
(2)若
,且集合
具有性质P,求x的值;
(3)若X具有性质P,且
,q为常数且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f7a53ccddc5210a37f12e3ab6e99df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3fe482c5e20abfc9f89c876f653ae3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/966888395e433b9c2a30138e7fb59cb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c122d308af408739c2717376e932122d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37c6bb4424eb1e5ab02b8ac83fd6ad10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8de3dabcc3150fd539ac97718ba10c5.png)
(1)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66317f3834697e2b5642906bb751eb25.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b7511e6ce72a5232820b7007f976be9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/864dd49f786346bc320deace92f949b0.png)
(3)若X具有性质P,且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2f5028bb9e126607ef62b402300c1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eda6dc559d07bc22c9a0ed1e3a6d01d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57313119f26fc9ba177f6ce7b57ab4f3.png)
您最近一年使用:0次
2024-04-23更新
|
315次组卷
|
2卷引用:江苏省南京市金陵中学2023-2024学年高一下学期第一次(3月)学情调研测试数学试题
名校
4 . 给出集合
对任意
,都有
成立
.
(1)若
,求证:函数
;
(2)由于(1)中函数
既是周期函数又是偶函数,于是张同学猜想了两个结论:
命题甲:集合
中的元素都是周期为6的函数;
命题乙:集合
中的元素都是偶函数;
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305c81b6a05c983ef0dd04962d546bd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4166972dec0aa3e8694a44eeb941a08.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b005e1e4b8e41c0028cd464835c464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4ca8bdc812627d925f00ed7c145d696.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5314a9d2205a2beba0dcffb8fd943b18.png)
(2)由于(1)中函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71d6c8ce1327c39675b26deeb0cfa49c.png)
命题甲:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
命题乙:集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
请对两个命题给出判断,如果正确,请证明;如果不正确,请举反例
您最近一年使用:0次
5 . 对于正整数集合
(
),如果任意去掉其中一个元素![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
之后,剩余的所有元素组成的集合都能分为两个交集为空集的集合,且这两个集合的所有元素之和相等,就称集合A为“可分集合”;
(1)判断集合
和
是否是“可分集合”(不必写过程);
(2)求证:四个元素的集合
一定不是“可分集合”;
(3)若集合
是“可分集合”,证明:
为奇数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3a3f24673b6e954db3a8b229d8c4564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7694f1219e3a480e81f62b29915b03d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50a272adba0f1120109824440f0e252c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cecc3d59296521ff4e1edc78a4ea67d7.png)
(1)判断集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e9859aa908844a32c0e1e069a046727.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a44d462b5c1b7b7ea6c0f36e5cab65b9.png)
(2)求证:四个元素的集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a784e0ba1c17aba6990123fe39b89114.png)
(3)若集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffbfa3e226e067ec597ebf0bbc2e87d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
您最近一年使用:0次
名校
解题方法
6 . 对于数集
,其中
,
,定义向量集
,若对任意
,存在
,使得
,则称
具有性质
.
(1)已知数集
,请写出数集
对应的向量集
,并判断
是否具有性质
(不需要证明).
(2)若
,且
具有性质
,求
的值;
(3)若
具有性质
,且
,
为常数且
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f7a53ccddc5210a37f12e3ab6e99df2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d3fe482c5e20abfc9f89c876f653ae3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c5751a1b2fb31063f3360f4ef5b0274.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c122d308af408739c2717376e932122d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37c6bb4424eb1e5ab02b8ac83fd6ad10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8de3dabcc3150fd539ac97718ba10c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(1)已知数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c40ffb95d55e922a408458c19940dbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/351bb3f3c54604330fa5b6c2bc3a7502.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/096b1ece1dcd29c59a46a4b3e02cb548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17bda892497cea43df67db57b4e2a07a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73eecc365f7e94267552eb430f2034e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)若
![](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/6734531288c894a5edb143104e448ce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7b68031c3405c23f82fb3f352e44a04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eaa18be2cbabe89d886b99241c4dca28.png)
您最近一年使用:0次
7 . 设自然数
,由
个不同正整数
构成集合
,若集合
的每一个非空子集所含元素的和构成新的集合
,记
为集合
元素的个数
(1)已知集合
,集合
,分别求解
.
(2)对于集合
,若
取得最大值,则称该集合
为“极异集合”
①求
的最大值(无需证明).
②已知集合
是极异集合,记
求证:数列
的前
项和
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bcfc48f9bc23cc43085bdb910e7a136.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c23747e7321187323c665a641adb49e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e29fd31a1808968790032a671f64be90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e29fd31a1808968790032a671f64be90.png)
(1)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ccaf946a034215b8c49c12a1aff7790.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d160c80e7542650f9ac8ff3981548ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11391d21b5da91adc137d57a73c19b83.png)
(2)对于集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9a84d5316c5db87fda9b7d615c9dce6.png)
②已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84357e7e15c0b4187ea69a4d555ef171.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35e9ca7e90d47d7ee295338bbac2d8d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b783cf91e34e692ce8e171f0965cb53f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7477e3d7c54f409ee9905e81c9cbe2f.png)
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8 . 对任意正整数n,记集合
,
.
,
,若对任意
都有
,则记
.
(1)写出集合
和
;
(2)证明:对任意
,存在
,使得
;
(3)设集合
.求证:
中的元素个数是完全平方数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a39352d44787ecda055946f530893f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5359f5f086b89cc656cdd4f79a3b7baa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5df0913f90131b298c8f6f57437f69b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c1352dca7dc3caf67c1cb937d52795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d35755ad07f05e7bfe00176d6334389f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75a28908216e6879a09b372d957be1e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(1)写出集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd3b9e816b14051f785aa5aae72b8eed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43a71fc9c0068109dad1382354570665.png)
(2)证明:对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e840ba7606959ccea36793f3ef0775d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b576952835af1f3492f0f3e6d00093e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90df641ab645927ee577e79faf18dcdd.png)
(3)设集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/112d102cbe7ce2bebe0c76e87e89a00c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08eb71ecf8d733b6932f4680874dbbf3.png)
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2023-11-15更新
|
165次组卷
|
4卷引用:北京市朝阳区2022届高三上学期期末统一检测数学试题
名校
9 . 设A为非空集合,令
,则
的任意子集R都叫做从A到A的一个关系(Relation),简称A上的关系.例如
时,
{0,2},![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76986e6f96cbb9d7d6d0fbcf0bf2321a.png)
,
,
{(0,0),(2,1)}等都是A上的关系.设R为非空集合A上的关系.给出如下定义:
①(自反性)若
,有
,则称R在A上是自反的;
②(对称性)若
,有
,则称R在A上是对称的;
③(传递性)若
,有
,则称R在A上是传递的;
如果R同时满足这3条性质,则称R为A上的等价关系.
(1)已知
,按要求填空:
①用列举法写出
______________________;
②A上的关系有____________个(用数值做答);
③用列举法写出A上的所有等价关系:{(0,0),(1,1),(2,2)},{(0,0),(1,1),(2,2),(0,1),(1,0)},{(0,0),(1,1),(2,2),(0,2),(2,0)},_______________,_______________,共5个.
(2)设
和
是某个非空集合A上的关系,证明:
①若
,
是自反的和对称的,则
也是自反的和对称的;
②若
,
是传递的,则
也是传递的.
(3)若给定的集合A有n个元素(
),
,
,...,
为A的非空子集,满足
且两两交集为空集.求证:
为A上的等价关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff994543fe18b563c7127c8b2a874358.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95c06ed271ff0a6407a3bf5deec5871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5689e50a9353ba69ff5b71e7b6a3c795.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8225d0531fba46cbb4a3af4dd2d6751f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76986e6f96cbb9d7d6d0fbcf0bf2321a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95c06ed271ff0a6407a3bf5deec5871.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934909fce1b90557163c6f43d4f0790d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8225d0531fba46cbb4a3af4dd2d6751f.png)
①(自反性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd170c506a8ce70f550f5751ae016ca6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e96fb327d44b08d715e86db04cc9785.png)
②(对称性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/328ae22ec119ce8f0faac8dc554a2c10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c950781f08495bc2a4c20454c26c48d8.png)
③(传递性)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/227539cbcd96eb67cbcf7c94de56598d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78dd7c7d34bcfcae1f423a684aae9542.png)
如果R同时满足这3条性质,则称R为A上的等价关系.
(1)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5689e50a9353ba69ff5b71e7b6a3c795.png)
①用列举法写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b0949c177d28fe5b6ec4a0de58c80a.png)
②A上的关系有____________个(用数值做答);
③用列举法写出A上的所有等价关系:{(0,0),(1,1),(2,2)},{(0,0),(1,1),(2,2),(0,1),(1,0)},{(0,0),(1,1),(2,2),(0,2),(2,0)},_______________,_______________,共5个.
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05725d20ff805152beff52c7a5e8d735.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efc18a5bb2e53586331b2a58538a48b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19f20f21a9d50b61dac519a3ddab539d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7077a5e7dce0e2f0e678b1147deae46.png)
(3)若给定的集合A有n个元素(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5818ede14d21f6df9ef9c2bfe09286c.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/7f4bae4bf0e8cf84b9e1c6c7258b06d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79ff32c9e80fd90fcdb360f9a5a21c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/591eaeea196d5720d0762ced03e8ce3b.png)
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名校
10 . 已知M是由满足下述条件的函数构成的集合:对任意
,①方程
有实数根;②函数
的导数
满足
.
(1)判断函数
是集合M中的元素,并说明理由;
(2)集合M中的元素
具有下面的性质:若
的定义域为D,则对于任意
,都存在
,使得等式
成立.试用这一性质证明:方程
有且只有一个实数根;
(3)对任意
,且
,求证:对于
定义域中任意的
,
,
,当
,且
时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57cac663990f61a4a3086c6bea3d51f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd689fbacfbe6c1bd0953521bbf3638b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/724340d69477c0ec2418c392b22b1cab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf4b85c070b9511e29c2677728f425c5.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1994b5f36e31a7d78bea39f6e71fcfc7.png)
(2)集合M中的元素
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5313c921defe84689aefde4773ad2b56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26597a94b1f024b0f75bb1bd76ccc4bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14d0efbfaf649d6ff75ad91f3a69b423.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd689fbacfbe6c1bd0953521bbf3638b.png)
(3)对任意
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57cac663990f61a4a3086c6bea3d51f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85b7e150af2052a1664cde963273d905.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09f86f37ec8e15846bd731ab4fcdbacd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec19b68e3add9d5bfcc6269a1855b87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/291c25fc6a69d6d0ccfb8d839b9b4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d42367830867a96861460aebfd53a0e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a385a2c08126b919ebf35e4121595842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc10d2a4a6a6d6cb8014cd7312f16053.png)
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