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