{"title":"注入集合和射出集","authors":"Natthajak Kamkru, Nattapon Sonpanow","doi":"10.1002/malq.202300059","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$\\mathfrak {m}$</annotation>\n </semantics></math>, denoted by <span></span><math>\n <semantics>\n <mrow>\n <mi>I</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$I(\\mathfrak {m})$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>J</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$J(\\mathfrak {m})$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msup>\n <mi>m</mi>\n <mi>m</mi>\n </msup>\n <annotation>$\\mathfrak {m}^\\mathfrak {m}$</annotation>\n </semantics></math>, respectively. Among our results, we show that “<span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mo>seq</mo>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo>≠</mo>\n <mi>I</mi>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo>≠</mo>\n <mo>seq</mo>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\operatorname{seq}^{1-1}(\\mathfrak {m})\\ne I(\\mathfrak {m})\\ne \\operatorname{seq}(\\mathfrak {m})$</annotation>\n </semantics></math>”, “<span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mo>seq</mo>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo>≠</mo>\n <mi>J</mi>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo>≠</mo>\n <mo>seq</mo>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\operatorname{seq}^{1-1}(\\mathfrak {m})\\ne J(\\mathfrak {m})\\ne \\operatorname{seq}(\\mathfrak {m})$</annotation>\n </semantics></math>” and “<span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mo>seq</mo>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <mo><</mo>\n <msup>\n <mi>m</mi>\n <mi>m</mi>\n </msup>\n <mo>≠</mo>\n <mo>seq</mo>\n <mrow>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\operatorname{seq}^{1-1}(\\mathfrak {m})&lt;\\mathfrak {m}^\\mathfrak {m}\\ne \\operatorname{seq}(\\mathfrak {m})$</annotation>\n </semantics></math>” are provable for an arbitrary infinite cardinal <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$\\mathfrak {m}$</annotation>\n </semantics></math>, and these are the best possible results, in the Zermelo-Fraenkel set theory (<span></span><math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math>) without the Axiom of Choice. Also, we show that it is relatively consistent with <span></span><math>\n <semantics>\n <mi>ZF</mi>\n <annotation>$\\mathsf {ZF}$</annotation>\n </semantics></math> that there exists an infinite cardinal <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$\\mathfrak {m}$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>I</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n <mo><</mo>\n <mi>J</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$S(\\mathfrak {m})=I(\\mathfrak {m})&lt;J(\\mathfrak {m})$</annotation>\n </semantics></math> where <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mo>(</mo>\n <mi>m</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$S(\\mathfrak {m})$</annotation>\n </semantics></math> denotes the cardinality of the set of bijections on a set which is of cardinality <span></span><math>\n <semantics>\n <mi>m</mi>\n <annotation>$\\mathfrak {m}$</annotation>\n </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"275-285"},"PeriodicalIF":0.4000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The set of injections and the set of surjections on a set\",\"authors\":\"Natthajak Kamkru, Nattapon Sonpanow\",\"doi\":\"10.1002/malq.202300059\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$\\\\mathfrak {m}$</annotation>\\n </semantics></math>, denoted by <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>I</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$I(\\\\mathfrak {m})$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>J</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$J(\\\\mathfrak {m})$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <msup>\\n <mi>m</mi>\\n <mi>m</mi>\\n </msup>\\n <annotation>$\\\\mathfrak {m}^\\\\mathfrak {m}$</annotation>\\n </semantics></math>, respectively. Among our results, we show that “<span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mo>seq</mo>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>≠</mo>\\n <mi>I</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>≠</mo>\\n <mo>seq</mo>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\operatorname{seq}^{1-1}(\\\\mathfrak {m})\\\\ne I(\\\\mathfrak {m})\\\\ne \\\\operatorname{seq}(\\\\mathfrak {m})$</annotation>\\n </semantics></math>”, “<span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mo>seq</mo>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>≠</mo>\\n <mi>J</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>≠</mo>\\n <mo>seq</mo>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\operatorname{seq}^{1-1}(\\\\mathfrak {m})\\\\ne J(\\\\mathfrak {m})\\\\ne \\\\operatorname{seq}(\\\\mathfrak {m})$</annotation>\\n </semantics></math>” and “<span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mo>seq</mo>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo><</mo>\\n <msup>\\n <mi>m</mi>\\n <mi>m</mi>\\n </msup>\\n <mo>≠</mo>\\n <mo>seq</mo>\\n <mrow>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\operatorname{seq}^{1-1}(\\\\mathfrak {m})&lt;\\\\mathfrak {m}^\\\\mathfrak {m}\\\\ne \\\\operatorname{seq}(\\\\mathfrak {m})$</annotation>\\n </semantics></math>” are provable for an arbitrary infinite cardinal <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$\\\\mathfrak {m}$</annotation>\\n </semantics></math>, and these are the best possible results, in the Zermelo-Fraenkel set theory (<span></span><math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math>) without the Axiom of Choice. Also, we show that it is relatively consistent with <span></span><math>\\n <semantics>\\n <mi>ZF</mi>\\n <annotation>$\\\\mathsf {ZF}$</annotation>\\n </semantics></math> that there exists an infinite cardinal <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$\\\\mathfrak {m}$</annotation>\\n </semantics></math> such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>I</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n <mo><</mo>\\n <mi>J</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$S(\\\\mathfrak {m})=I(\\\\mathfrak {m})&lt;J(\\\\mathfrak {m})$</annotation>\\n </semantics></math> where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n <mo>(</mo>\\n <mi>m</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$S(\\\\mathfrak {m})$</annotation>\\n </semantics></math> denotes the cardinality of the set of bijections on a set which is of cardinality <span></span><math>\\n <semantics>\\n <mi>m</mi>\\n <annotation>$\\\\mathfrak {m}$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49864,\"journal\":{\"name\":\"Mathematical Logic Quarterly\",\"volume\":\"70 3\",\"pages\":\"275-285\"},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Logic Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300059\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300059","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们研究了注入集合、射出集合和一个集合上所有函数的集合的心数之间的关系，这些集合的心数为 m $\mathfrak {m}$，分别用 I ( m ) $I(\mathfrak {m})$、J ( m ) $J(\mathfrak {m})$和 m m $\mathfrak {m}^\mathfrak {m}$表示。Among our results, we show that “ seq 1 − 1 ( m ) ≠ I ( m ) ≠ seq ( m ) $\operatorname{seq}^{1-1}(\mathfrak {m})\ne I(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$ ”, “ seq 1 − 1 ( m ) ≠ J ( m ) ≠ seq ( m ) $\operatorname{seq}^{1-1}(\mathfrak {m})\ne J(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$ ” and “ seq 1 − 1 ( m ) < m m ≠ seq ( m ) $\operatorname{seq}^{1-1}(\mathfrak {m})<\mathfrak {m}^\mathfrak {m}\ne \operatorname{seq}(\mathfrak {m})$ ” are provable for an arbitrary infinite cardinal m $\mathfrak {m}$ , and these are the best possible results, in the Zermelo-Fraenkel set theory ( ZF $\mathsf {ZF}$ ) without the Axiom of Choice.另外，我们还证明，存在一个无限红心 m $\mathfrak {m}$，使得 S ( m ) = I ( m ) <； J ( m ) $S(\mathfrak {m})=I(\mathfrak {m})<J(\mathfrak {m})$ 其中 S ( m ) $S(\mathfrak {m})$表示一个集合上双射集合的万有性，这个集合的万有性为 m $\mathfrak {m}$ 。
The set of injections and the set of surjections on a set
In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality , denoted by , and , respectively. Among our results, we show that “”, “” and “” are provable for an arbitrary infinite cardinal , and these are the best possible results, in the Zermelo-Fraenkel set theory () without the Axiom of Choice. Also, we show that it is relatively consistent with that there exists an infinite cardinal such that where denotes the cardinality of the set of bijections on a set which is of cardinality .
期刊介绍：
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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