Problem related with a set

Question

I was thinking about the following problem:

Let $X$ denote the two point $\{0,1\}$ and write $X_{j}=\{0,1\}$ for every j=1,2,3,....Let $Y=\prod_{j=1}^{\infty}X_{j}.$ Then, which of the following is/are true?

1. $Y$ is a countable set,
2. Card $Y$=card$[0,1],$
3. $\bigcup_{n=1}^{\infty}$ ($\prod_{j=1}^{n}X_{j})$ is uncountable,
4. $Y$ is uncountable.

Please help.Thanks in advance for your time.

Answer 1

*Hints:*$\newcommand{\card}{\operatorname{card}}$

1. $\card(\mathcal P(\mathbb N))=\card(\prod_{n=1}^\infty\{0,1\})$.
2. For every set $A$, $\card(A)<\card(\mathcal P(A))$.
3. Finite products of finite sets are finite. Countable unions of finite sets are countable.
4. $\card([0,1])=\card(\mathbb R)=\card(\mathcal P(\mathbb N))$.