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

• Published in: Mathematical logic quarterly Vol. 70; no. 3; pp. 275 - 285
• Main Authors: Kamkru, Natthajak, Sonpanow, Nattapon
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.08.2024
• Subjects: Set theory
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202300059

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$\mathfrak {m}$, denoted by I(m)$I(\mathfrak {m})$, J(m)$J(\mathfrak {m})$ and mm$\mathfrak {m}^\mathfrak {m}$, respectively. Among our results, we show that “seq1−1(m)≠I(m)≠seq(m)$\operatorname{seq}^{1-1}(\mathfrak {m})\ne I(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$”, “seq1−1(m)≠J(m)≠seq(m)$\operatorname{seq}^{1-1}(\mathfrak {m})\ne J(\mathfrak {m})\ne \operatorname{seq}(\mathfrak {m})$” and “seq1−1(m)<mm≠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. Also, we show that it is relatively consistent with ZF$\mathsf {ZF}$ that there exists an infinite cardinal m$\mathfrak {m}$ such that S(m)=I(m)<J(m)$S(\mathfrak {m})=I(\mathfrak {m})<J(\mathfrak {m})$ where S(m)$S(\mathfrak {m})$ denotes the cardinality of the set of bijections on a set which is of cardinality m$\mathfrak {m}$.