The set of injections and the set of surjections on a set

Pub Date : 2024-09-16 DOI:10.1002/malq.202300059

Natthajak Kamkru, Nattapon Sonpanow

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Abstract

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 $m$ , denoted by $I (m)$ , $J (m)$ and $m^{m}$ , respectively. Among our results, we show that “ ${seq}^{1 - 1} (m) \neq I (m) \neq seq (m)$ ”, “ ${seq}^{1 - 1} (m) \neq J (m) \neq seq (m)$ ” and “ ${seq}^{1 - 1} (m) < m^{m} \neq seq (m)$ ” are provable for an arbitrary infinite cardinal $m$ , and these are the best possible results, in the Zermelo-Fraenkel set theory ( $ZF$ ) without the Axiom of Choice. Also, we show that it is relatively consistent with $ZF$ that there exists an infinite cardinal $m$ such that $S (m) = I (m) < J (m)$ where $S (m)$ denotes the cardinality of the set of bijections on a set which is of cardinality $m$ .

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注入集合和射出集

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 $\mathfrak {m}$ 。

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