Please help me with these questions! Any ideas or hints or anything would be great. We've had exactly one lecture on this, and the book has no proofs, so I'm struggling with how to formulate proofs for this type of thing. Here are the problems and my ideas so far:
1. Show that for real numbers and with has the same cardinality of .
I know this interval is uncountably infinite like , and I can use Cantor's diagonalization argument to show that. Is this all I need to do?
2. Suppose and are sets such that card card . Prove there exists a set such that card = card .
I'm trying to use the regular existence proof idea, in the sense that I'm trying to think of a "Consider this set C and the bijection f:A to C" and so on, but I can't think of anything that applies to an abstract set. A could have its elements as numbers, functions, subsets, etc. and all the things I can think of require something to be known about the set.
3. Suppose that , and are sets such that card < card and card card . Prove that card < card .
I'm totally lost on this one... It seems obvious, so I think I'm having trouble separating the properties of finite sets with the properties of infinite sets.
Each of these three questions is just an exercise is applying the definitions.
Each involves finding a function that applies.
In #1 can you give a bijection between and ?
Think tangent and arctangent.
What is the meaning of card(A)<card(B)? The image of a subset of A is a subset of B.
What do you know about the composition of functions?