# Empty Families

1. Jun 21, 2009

### doktordave

I am beginning to study set theory and came across the following example:

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

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 $$\emptyset$$? And since the empty set is defined not to contain anything, how could it contain any subsets of the set of real numbers?

2. Jun 21, 2009

### slider142

It does not contain anything. The second sentence is vacuously true.

3. Jun 21, 2009

### Preno

Actually, the intersection of the empty set is V, the class of all sets.

4. Jun 21, 2009

### HallsofIvy

Staff Emeritus
Yes, that's true. "every member of $$\mathcal{A}$$ contains all real numbers" is the same as "if U is a member of $$\mathcal{A}$$ then U contains all real numbers". The statement "if A then B" is true whenever A is false, irrespective of whether B is true or false (that is what slider142 means by "vacuously true"). Since "U is a member of $$\mathcal{a}$$ is always false, anything we say about U is true!

It doesn't. That is not what the statement says!

5. Jun 21, 2009

### doktordave

I think I understand now. Since the intersection over $$\mathcal{A}$$ is defined as $$\left\{x: (\forall A)(A\in \mathcal{A} \Rightarrow x\in A)\right\}$$ and the antecedent of the conditional is always false (there is nothing in $$\mathcal{A}$$), the conditional will always be true, because of the way the conditional operator is defined. So x can be anything in the universe. This seems a little backwards to my way of thinking, but I guess that's ok. I'll have to study that article on vacuous truth, it looks interesting. Thanks!

edit: Ah, thanks HallsofIvy. I was busy editing this post while you responded.