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Abstract: Let if , and if . We conjecture that the θ-orbit of every nonnegative rational number ends in 0. A weaker conjecture asserts that there are no positive rational fixed points for any map in the semigroup Λ generated by the maps and . In this paper, we prove that the asymptotic density of the set of maps in Λ that have rational fixed points is zero. Moreover, we prove that certain types of elements in the semigroup Λ cannot have rational fixed points. [Copyright &y& Elsevier]

Abstract: A system of distinct representatives (SDR) of a family is a sequence of distinct elements with for . Let denote the number of SDRs of a family ; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer , a family is called a -family if the union of any sets in the family contains at least elements. The famous Hall’s theorem says that if and only if is a -family. Denote by the minimum number of SDRs in a -family. The problem of determining and those families containing exactly SDRs was first raised by Chang [G.J. Chang, On the number of SDR of a -family, European J. Combin. 10 (1989) 231–234]. He solved the cases when and gave a conjecture for . In this paper, we solve the conjecture. [Copyright &y& Elsevier]