Affiliation:
1. Department of Mathematics and Statistics, Faculty of Science and Technology Thammasat University Thailand
2. Department of Mathematics and Computer Science, Faculty of Science Chulalongkorn University Thailand
Abstract
AbstractWe write and for the cardinalities of the set of permutations with n non‐fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality , where n is a natural number greater than 1. With the Axiom of Choice, and are equal for all infinite cardinals . We show, in ZF, that if is assumed, then for any infinite cardinal . Moreover, the assumption cannot be removed for and the superscript cannot be replaced by n. We also show under that for any infinite cardinal , implies is Dedekind‐infinite.