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.