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Let N be a set of N elements and F 1 , F 2 ,... be a sequence of random independent equiprobable mappings N → N. For a subset S 0  ⊂ N, |S 0 | = n, we consider a sequence of its images S k  = F k (. . . F 2 (F 1 (S 0 )) . . .), k = 1, 2... , and a sequence of their unions Ψ k  = S 1  ⋃ ... ⋃ S k , k = 1, 2 ... An approach to the exact computation of distribution of |S k | and |Ψ k | for moderate values of N is described. Two-sided inequalities for M|S k | and M|Ψ k | such that upper bound are asymptotically equivalent to lower ones for N, n, k → ∞, nk = o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

Images of subset of finite set under iterations of random mappings