I'm so confused about cardinalities of some sets. What is the countable infinite product of a two points set $\{0,1\}$? Does it have the same cardinality as the real number $\mathbb R$? Or is the infinite product just countable?
Could anyone give me the answer?
On
Let $X$ be a set and consider the set $\{0,1\}^X$ of all maps $X\to\{0,1\}$. Then there is a bijection $$ f\colon \mathcal{P}(X)\to\{0,1\}^X $$ ($\mathcal{P}(X)$ is the power set of $X$) defined by sending each subset $A$ of $X$ to its characteristic function $$ \chi_A(x)=\begin{cases} 1 & \text{if $x\in A$}\\ 0 & \text{if $x\notin A$} \end{cases} $$ Thus $\{0,1\}^X$ has the same cardinality of $\mathcal{P}(X)$.
The countably infinite product of the set $\{0,1\}$ is simply the set of all infinite binary strings which Cantor showed to be uncountable in the classic diagonalization argument.