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

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 , 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 .