  In the several contexts such as combinatorial number theory, families of sets
of positive integers closed under taking subsets have been investigated. %To
investigate some property of an infinite set of positive integers, Then it is
sometimes useful to give bijections between the set of the one-sided infinite
sequences on the alphabet set $\{0,1\}$ and such a family of sets. The most
typical example is the family of sum-free sets. Although such a kind of
families covers a large class of families of sets, there are only a few
considerations on bijections for the case where the sum-free property is
replaced by another property.
  In this paper, we explicitly give a bijection and its inverse between the set
of one-sided infinite sequences on the alphabet set $\{0,1\}$ and a family of
sets which may be contained in a class of families closed under taking subsets.
Moreover, we show that some extremal property in a particular family of sets is
characterized by a discrete dynamical system based on this kind of bijections.
