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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.

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Do you mean $Y_j$? Or just "any" of the products? You wrote an infinite list of products assigned to a single variable. –  Asaf Karagila Dec 13 '12 at 17:38
    
How are you using $X$? What does $X_j=0,1$ mean, is it $0,1$ or $\{0,1\}$? –  gt6989b Dec 13 '12 at 17:39
    
I am sorry. i am editing now. –  user52976 Dec 13 '12 at 17:40

1 Answer 1

up vote 2 down vote accepted

*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. $\card([0,1])=\card(\mathbb R)=\card(\mathcal P(\mathbb N))$.
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