Consider the following problem:
Which of the following sets has the greatest cardinality?
A. ${\mathbb R}$
B. The set of all functions from ${\mathbb Z}$ to ${\mathbb Z}$
C. The set of all functions from ${\mathbb R}$ to $\{0,1\}$
D. The set of all finite subsets of ${\mathbb R}$
E. The set of all polynomials with coefficients in ${\mathbb R}$
What I can get is that $\#(A)=2^{\aleph_0}$ and $\#(C)=2^{2^{\aleph_0}}.$ And I think $\#(D)=\#(E)$. For B, one may get $\aleph_0^{\aleph_0}$. But how should I compare it with others(especially C)?
Here is my question:
What are cardinalities for B, D and E?