I am beginning to study set theory and came across the following example:(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]\mathcal{A}[/tex] be the empty family of subsets of [tex]\mathbb{R}[/tex]. Since [tex]\mathcal{A}[/tex] is empty, every member of [tex]\mathcal{A}[/tex] contains all real numbers. That is, [tex]((\forall A)(A\in\mathcal{A}\Rightarrow x\in A))[/tex] is true for all real numbers x. Thus [tex]\bigcap_{A\in\mathcal{A}} A = \mathbb{R}[/tex].

My problem is with the first sentence. Since a family is simply a set of sets, If we talk about an empty family wouldn't this simply be the empty set [tex]\emptyset[/tex]? And since the empty set is defined not to contain anything, how could it contain any subsets of the set of real numbers?

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# Empty Families

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