We define by recursion the set of sets {An:n∈ℕ} this way:(adsbygoogle = window.adsbygoogle || []).push({});

A_0=∅

A_n+1=A_n ∪ {A_n}.

I want to prove by induction that for all n∈ℕ, the set A_n has n elements and that A_n is transitive (i.e. if x∈y∈A_n, then x∈A_n).

My thoughts:

for n=0, A_1 = ∅∪ {∅} = {∅}

then, for n+1: A_n+2 =A_n+1 ∪ {A_n+1}

I'm confused on how to proceed.

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# Show that the set of sets {An} has n elements

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