(a) I can explain the premise behind constructing such a bijection - it is up to you to actually write it up, of course.

First, a little background on power sets. For any set

with a cardinality of

, its power set

has the same cardinality as

The 0's and 1's are indicator functions telling you whether or not an element of

in a particular subset or not.

For example, consider the set

, which a cardinality of 3 and a power set

which is

empty set

Then,

Do you see how the set

empty set

corresponds to

,

to

, so forth?

Extending this principle to infinite sets is possible, so we conclude that

has the same cardinality as

(multiplied together countable times). But what is this set? This set is all the possible countably sequences of 0's and 1's that exist out there.

Note that if you convert any given real number into binary, you get the same thing: sequences of 0's and 1's. So that's your bijection.