I'm stuck on the following question:
Let A be the set of all real-valued functions on [0,1]. show that there does nor exist a function from [0,1] onto A.
I don't even know where to start.
This is a rather advanced theorem.
So I have no way of knowing what you have already proved.
But here are the lemmas you will need.
1) There is an injection if and only if there is a surjection .
2) There is no surjection , (power set of ).
3) If , the set of functions from having only two values: 0 or 1.
Then
Now clearly the set
If there were then
.