Entwining

Probability Level 3

True or false? :

"Given $n$ countably infinite sets $A_1, A_2,A_3,\ldots,A_n$ with $n\geq2$ , there exist a positive integer $m\leq n$ such that the Cartesian product $A_1 \times A_2 \times \ldots \times A_m$ is uncountably infinite."

False This question makes no sense True Like the continuum hypothesis, we can't decide whether it's true or false

1 solution

Otto Bretscher
May 22, 2015

"The Cartesian product of two countable sets is countable."

It suffices to show that $\mathbb{N}\times\mathbb{N}$ is countable, meaning that there exists an injection $f$ from $\mathbb{N}\times\mathbb{N}$ to $\mathbb{N}$ . Indeed, we have $f(m,n)=2^m3^n$ .

Entwining

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