Let n be a natural number. Show that if ℱ is a finite collection of n sets, then ⟨ℱ⟩bool is a f

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Let $n$ be a natural number. Show that if $\mathcal{F}$ is a finite collection of $n$ sets, then $\langle\mathcal{F}\rangle_{\text {bool }}$ is a finite Boolean algebra of cardinality at most $2^{2^n}$ (in particular, finite sets generate finite algebras). Give an example to show that this bound is best possible. (Hint: for the latter, it may be convenient to use a discrete ambient space such as the discrete cube $X=\{0,1\}^n$.)

    Let $n$ be a natural number. Show that if $\mathcal{F}$ is a finite collection of $n$ sets, then $\langle\mathcal{F}\rangle_{\text {bool }}$ is a finite Boolean algebra of cardinality at most $2^{2^n}$ (in particular, finite sets generate finite algebras). Give an example to show that this bound is best possible. (Hint: for the latter, it may be convenient to use a discrete ambient space such as the discrete cube $X=\{0,1\}^n$.)

An Introduction To Measure Theory (January 2011 Draft)

Terence Tao 1st Edition

Chapter 1, Problem 8

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Let $n$ be a natural number. Show that if $\mathcal{F}$ is a finite collection of $n$ sets, then $\langle\mathcal{F}\rangle_{\text {bool }}$ is a finite Boolean algebra of cardinality at most $2^{2^n}$ (in particular, finite sets generate finite algebras). Give an example to show that this bound is best possible. (Hint: for the latter, it may be convenient to use a discrete ambient space such as the discrete cube $X=\{0,1\}^n$.)

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