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THE PERMUTATIONS WITH n NON-FIXED POINTS AND THE SEQUENCES WITH LENGTH n OF A SET
We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$ . With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm...
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| Published in: | The Journal of symbolic logic 2024-09, Vol.89 (3), p.1067-1076 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c229t-f1e4dc66cb4ef1b0709e1f1b9446b6c3793a5e276898c029c0662b956b258c4c3 |
| container_end_page | 1076 |
| container_issue | 3 |
| container_start_page | 1067 |
| container_title | The Journal of symbolic logic |
| container_volume | 89 |
| creator | NUNTASRI, JUKKRID VEJJAJIVA, PIMPEN |
| description | We write
$\mathcal {S}_n(A)$
for the set of permutations of a set A with n non-fixed points and
$\mathrm {{seq}}^{1-1}_n(A)$
for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than
$1$
. With the Axiom of Choice,
$|\mathcal {S}_n(A)|$
and
$|\mathrm {{seq}}^{1-1}_n(A)|$
are equal for all infinite sets A. Among our results, we show, in ZF, that
$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$
for any infinite set A if
${\mathrm {AC}}_{\leq n}$
is assumed and this assumption cannot be removed. In the other direction, we show that
$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A and the subscript
$n+1$
cannot be reduced to n. Moreover, we also show that “
$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A” is not provable in ZF. |
| doi_str_mv | 10.1017/jsl.2022.57 |
| format | article |
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$\mathcal {S}_n(A)$
for the set of permutations of a set A with n non-fixed points and
$\mathrm {{seq}}^{1-1}_n(A)$
for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than
$1$
. With the Axiom of Choice,
$|\mathcal {S}_n(A)|$
and
$|\mathrm {{seq}}^{1-1}_n(A)|$
are equal for all infinite sets A. Among our results, we show, in ZF, that
$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$
for any infinite set A if
${\mathrm {AC}}_{\leq n}$
is assumed and this assumption cannot be removed. In the other direction, we show that
$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A and the subscript
$n+1$
cannot be reduced to n. Moreover, we also show that “
$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A” is not provable in ZF.</description><identifier>ISSN: 0022-4812</identifier><identifier>EISSN: 1943-5886</identifier><identifier>DOI: 10.1017/jsl.2022.57</identifier><language>eng</language><publisher>New York, USA: Cambridge University Press</publisher><subject>Codes ; Theorems</subject><ispartof>The Journal of symbolic logic, 2024-09, Vol.89 (3), p.1067-1076</ispartof><rights>The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c229t-f1e4dc66cb4ef1b0709e1f1b9446b6c3793a5e276898c029c0662b956b258c4c3</citedby><cites>FETCH-LOGICAL-c229t-f1e4dc66cb4ef1b0709e1f1b9446b6c3793a5e276898c029c0662b956b258c4c3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.cambridge.org/core/product/identifier/S0022481222000573/type/journal_article$$EHTML$$P50$$Gcambridge$$H</linktohtml><link.rule.ids>314,780,784,27922,27923,72730</link.rule.ids></links><search><creatorcontrib>NUNTASRI, JUKKRID</creatorcontrib><creatorcontrib>VEJJAJIVA, PIMPEN</creatorcontrib><title>THE PERMUTATIONS WITH n NON-FIXED POINTS AND THE SEQUENCES WITH LENGTH n OF A SET</title><title>The Journal of symbolic logic</title><addtitle>J. symb. log</addtitle><description>We write
$\mathcal {S}_n(A)$
for the set of permutations of a set A with n non-fixed points and
$\mathrm {{seq}}^{1-1}_n(A)$
for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than
$1$
. With the Axiom of Choice,
$|\mathcal {S}_n(A)|$
and
$|\mathrm {{seq}}^{1-1}_n(A)|$
are equal for all infinite sets A. Among our results, we show, in ZF, that
$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$
for any infinite set A if
${\mathrm {AC}}_{\leq n}$
is assumed and this assumption cannot be removed. In the other direction, we show that
$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A and the subscript
$n+1$
cannot be reduced to n. Moreover, we also show that “
$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A” is not provable in ZF.</description><subject>Codes</subject><subject>Theorems</subject><issn>0022-4812</issn><issn>1943-5886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNptkE1PAjEQhhujiYie_ANNPJrFttvttscNdGET7IKU6K3ZLV0D4cstHPz3FiHx4mkmM8-8kzwAPGLUwwinLyu_7hFESC9Jr0AHCxpHCefsGnRQmEaUY3IL7rxfIYQSQXkHTPVIwol8e53rTBelmsH3Qo_gFqpSRXnxIQdwUhZKz2CmBvAEz-R0LlVfXsixVMPfgzKHWVjqe3DTVGvvHi61C-a51P1RNC6HRT8bR5YQcYga7OjCMmZr6hpcoxQJh0MjKGU1s3Eq4ipxJGVccIuIsIgxUouE1SThltq4C57Ouft293V0_mBWu2O7DS9NjImghKScBur5TNl2533rGrNvl5uq_TYYmZMzE5yZkzOTpIGOLnS1qdvl4tP9hf7H_wB6zmV6</recordid><startdate>202409</startdate><enddate>202409</enddate><creator>NUNTASRI, JUKKRID</creator><creator>VEJJAJIVA, PIMPEN</creator><general>Cambridge University Press</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>202409</creationdate><title>THE PERMUTATIONS WITH n NON-FIXED POINTS AND THE SEQUENCES WITH LENGTH n OF A SET</title><author>NUNTASRI, JUKKRID ; VEJJAJIVA, PIMPEN</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c229t-f1e4dc66cb4ef1b0709e1f1b9446b6c3793a5e276898c029c0662b956b258c4c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Codes</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>NUNTASRI, JUKKRID</creatorcontrib><creatorcontrib>VEJJAJIVA, PIMPEN</creatorcontrib><collection>CrossRef</collection><jtitle>The Journal of symbolic logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>NUNTASRI, JUKKRID</au><au>VEJJAJIVA, PIMPEN</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>THE PERMUTATIONS WITH n NON-FIXED POINTS AND THE SEQUENCES WITH LENGTH n OF A SET</atitle><jtitle>The Journal of symbolic logic</jtitle><addtitle>J. symb. log</addtitle><date>2024-09</date><risdate>2024</risdate><volume>89</volume><issue>3</issue><spage>1067</spage><epage>1076</epage><pages>1067-1076</pages><issn>0022-4812</issn><eissn>1943-5886</eissn><abstract>We write
$\mathcal {S}_n(A)$
for the set of permutations of a set A with n non-fixed points and
$\mathrm {{seq}}^{1-1}_n(A)$
for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than
$1$
. With the Axiom of Choice,
$|\mathcal {S}_n(A)|$
and
$|\mathrm {{seq}}^{1-1}_n(A)|$
are equal for all infinite sets A. Among our results, we show, in ZF, that
$|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$
for any infinite set A if
${\mathrm {AC}}_{\leq n}$
is assumed and this assumption cannot be removed. In the other direction, we show that
$|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A and the subscript
$n+1$
cannot be reduced to n. Moreover, we also show that “
$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$
for any infinite set A” is not provable in ZF.</abstract><cop>New York, USA</cop><pub>Cambridge University Press</pub><doi>10.1017/jsl.2022.57</doi><tpages>10</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0022-4812 |
| ispartof | The Journal of symbolic logic, 2024-09, Vol.89 (3), p.1067-1076 |
| issn | 0022-4812 1943-5886 |
| language | eng |
| recordid | cdi_proquest_journals_3129422784 |
| source | Cambridge University Press |
| subjects | Codes Theorems |
| title | THE PERMUTATIONS WITH n NON-FIXED POINTS AND THE SEQUENCES WITH LENGTH n OF A SET |
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