# Feedback on hw 7

## Cardinality of infinite sets

If A is an infinite set, one should be careful when talking about its cardinality |A|: what kind of object is it ? Whatever it is, we know that if e.g. A is the set of natural numbers, and B is the reals, |A| ≠ |B|, so we cannot just say that |A| and |B| are ∞ (whatever that means). It turns out one can see |A|, |B| as some kind of infinitary numbers, called cardinals, that will not be defined in this course. Fortunately, you can still talk about what it means for A and B to have the same cardinality using bijections, but unless A is finite, it is best to avoid writing |A| entirely, in my opinion.

We might revisit this subject once we get to equivalence relations.

## Countable sets are infinite

Your textbook defines a set as countable if it is in bijection with the natural numbers, thus if A is a countable set, one directly knows it is infinite (why ?).

## A surjection is enough to prove countability

When you want to *prove* a certain set A is countable (as in 4.49), it is enough to build a surjection from the natural into A, *and* prove that A is infinite.

The natural map many of you defined in 4.49 is *not* a bijection, as it could be that A_{1}, A_{2}, ... are not disjoint (e.g. they may all be equal), so the fact a surjection is enough comes in handy (but then you shouldn't forget to argue A is infinite).

## Finite vs infinite unions

A very serious mistake in 4.49 was to prove only that the union of A_{1}, ..., A_{k} is countable for all k. This is *not* the same as showing the entire union is countable. For example, if you replace "countable" by "finite", it is true that a union of finitely many finite sets is finite, but a countable union of finite sets is not necessarily finite (why ?). This shows that even if a property is true of any "initial segment" of a countable union, it may not hold for the entire union.