## Abstract

Let $\mathcal{N}$ be a set of *N* elements and $\left({F}_{1},{G}_{1}\right),\left({F}_{2},{G}_{2}\right),\dots $ be a sequence of independent pairs of random dependent mappings $\mathcal{N}\to \mathcal{N}$ such that *F*_{k} and *G*_{k} are random equiprobable mappings and $\mathbf{P}\{{F}_{k}(x)={G}_{k}(x)\}=\alpha $ for all $x\in \mathcal{N}$ and *k* = 1, 2, … For a subset ${S}_{0}\subset \mathcal{N},\phantom{\rule{0.056em}{0ex}}|{S}_{0}|=n$, we consider a sequences of its images ${S}_{k}={F}_{k}(\dots {F}_{2}({F}_{1}({S}_{0}))\dots )$, ${T}_{k}={G}_{k}(\dots {G}_{2}({G}_{1}({S}_{0}))\dots )$, *k* = 1, 2 …, and a sequences of their unions ${S}_{k}\cup {T}_{k}$ and intersections ${S}_{k}\cap {T}_{k}$, *k* = 1, 2 … We obtain two-sided inequalities for $\mathbf{M}|{S}_{k}\cup {T}_{k}|$ and $\mathbf{M}|{S}_{k}\cap {T}_{k}|$ such that upper and lower bounds are asymptotically equivalent if $N,n,k\to \mathrm{\infty}$, $nk=o(N)$ and $\alpha =O\left(\frac{1}{N}\right)$.

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