A collection of subsets of is called an **independent** **family** iff for any pairwise distinct in , we have that the set is infinite.

- Show that there is an independent family of the same size as the reals. (This means that there is an independent family for which there is a bijection between and . If it is easier to think of it this way, you can use that has the same size as the set of infinite sequences of zeros and ones.)
- Given a set recall that its
*characteristic function*is the map such that if and if We can think of each as a vector in the space Suppose now that is an independent family, and show that is a linearly independent set. (Thus any basis for over must have the same size as the reals.) - Consider as a vector space over and show that any basis must have the same size as the reals.