# The Intersection of Sets is a Set

## 2018 January 13

Modern mathematics is almost universally* based on set theory, so I decided to read Pinter's Set Theory.

Naive set theory says that a set is any collection of objects. For example, the set of natural numbers, , is the set {1, 2, 3, …}. Naive set theory runs into several problems, so most mathematicians use Zermelo-Fraenkel (ZF) set theory, so this post also uses it. The difference is not important for this post.

The intersection A ∩ B of 2 sets, A and B, is the set of objects which are in both A and B. For example,
{1, 2, 3}∩{2, 3, 4}={2, 3}

Note that the associative property holds for . Also,
$$\bigcap_{i \in I}^{} A_i = A_1 \cap A_2 \cap \ldots \cap A_n$$

Here is a proof of an exercise in Set Theory.

Conjecture

Let {Ai}i ∈ I be an indexed family of sets. Prove that
$$\bigcap_{i \in I}^{} A_i$$
is a set.

Proof

By definition, the intersection of 2 sets is a set. The associative property holds for the intersection, so
$$\bigcap_{i \in I}^{} A_i = A_1 \cap (A_2 \cap \ldots \cap A_n)$$
We already know that A1 is a set, so $\bigcap_{i \in I}^{} A_i$ is a set if (A2 ∩ … ∩ An) is a set. We can repeat this step, writing that
A2 ∩ … ∩ An = A2 ∩ (A3 ∩ …An)
Again, we see that A2 is a set, so we now need to prove A3 ∩ …An is a set. We repeat this step until we get
An − 1 ∩ (An)
which is a set, so now we go backwards, seeing that
An − 2 ∩ (An − 1 ∩ An)
is a set, and continuing backwards until we prove that
A1 ∩ (A2 ∩ … ∩ An)
is a set, which equals
$$\bigcap_{i \in I}^{} A_i$$
Thus, $\bigcap_{i \in I}^{} A_i$ is a set. QED

## Notes

<li id="lambda-category">Aside from set theory, there are several other
useful foundations for mathematics, such as lambda calculus and category
theory.  Lambda calculus is very popular in computer science in particular,
where it is the inspiration for functional programming using languages such
as Haskell.  Category theory is also really interesting because it's just
so abstract that even mathematicians (jokingly) call it <a
href="https://en.wikipedia.org/wiki/Abstract_nonsense">abstract
nonsense</a>.</li>