Originally Posted by

**dwsmith** Which of the following sets has the greatest cardinality?

All but (c) have the same cardinality $\displaystyle 2^{\aleph_0}=c$

(A) $\displaystyle \mathbb{R} $

A: $\displaystyle Card(\mathbb{R})=c$

(B) The set of all functions from $\displaystyle \mathbb{Z}\to\mathbb{Z}$

B: I think this is countable infinite $\displaystyle (\mathbb{Z}\to\mathbb{Z})\sim\mathbb{N}$

Its cardinality is, by definition, $\displaystyle |\mathbb{Z}|^{|\mathbb{Z}|}=\aleph_0^{\aleph_0}=c$

(C) The set of all functions from $\displaystyle \mathbb{R}\to${$\displaystyle 0,1$}

C: This is the answers but why?

Its cardinality is $\displaystyle |(0,1)|^{|\mathbb{R}|}=c^c>c$

(D) The set of all finite subsets of $\displaystyle \mathbb{R} $

D: Not sure how to determine cardinality here

For every $\displaystyle n\in\mathbb{N}\,\,\exists\,c=2^{\aleph_0}$ subsets of $\displaystyle \mathbb{R}$ with $\displaystyle n$ elements, so the set we're dealing with has cardinality $\displaystyle c\cdot \aleph_0=c$

(E) The set of all polynomials with coefficients in $\displaystyle \mathbb{R} $

E: I think this countable infinite too.