Abstract A partition is finitary if all its members are finite. For a set A , script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) denotes the set of all finitary partitions of A . It is shown consistent with upper Z upper F $\mathsf {ZF}$ Z F (without the axiom of choice) that there exist an infinite set A and a surjection from A onto script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) . On the other hand, we prove in upper Z upper F $\mathsf {ZF}$ Z F some theorems concerning script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) for infinite sets A , among which are the following: (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto script upper B left parenthesis upper A right parenthesis $\mathscr {B}(A)$ B ( A ) and no finite-to-one functions from script upper B left parenthesis upper A right parenthesis

Abstract It is proved in $\mathsf {ZF}$ (without the axiom of choice) that, for all infinite sets M, there are no surjections from $\omega \times M$ onto $\operatorname {\mathrm {\mathscr {P}}}(M)$ .

Abstract For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following: (1) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ , where ${{\cal S}_{{\rm{fin}}}}\left( x \right)$ denotes the set of all permutations of x which move only finitely many elements. (2) For all sets x such that ${\cal S}\left( x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ and there are no finite-to-one functions from ${\cal S}\left( x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x. (3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left( x \right)$ into x. (4) For all sets x, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$ .

Abstract For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$ , where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition. (2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$ . (3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.