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In the several contexts such as combinatorial number theory, families of setsof positive integers closed under taking subsets have been investigated. %Toinvestigate some property of an infinite set of positive integers, Then it issometimes useful to give bijections between the set of the one-sided infinitesequences on the alphabet set $\{0,1\}$ and such a family of sets. The mosttypical example is the family of sum-free sets. Although such a kind offamilies covers a large class of families of sets, there are only a fewconsiderations on bijections for the case where the sum-free property isreplaced by another property.  In this paper, we explicitly give a bijection and its inverse between the setof one-sided infinite sequences on the alphabet set $\{0,1\}$ and a family ofsets 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 ischaracterized by a discrete dynamical system based on this kind of bijections.

On the construction of a family of sets of positive integers closed under taking subsets