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?
Best Answer
The correct answer is the functions from $\mathbb R$ to $\{0,1\}$, the calculations and comparisons are given here:
In particular it means that the set of functions from $\mathbb R$ to $\{0,1\}$ is the largest, and in fact it is the only one not of cardinality continuum.