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Mark each statement as TRUE, FALSE, or UNKNOWN

(a) $|\Bbb{R}| < \aleph_1$

(b) $|\Bbb{R}| = \aleph_1$

(c) $|P(\Bbb{R})| > \aleph_1$

Could someone explain to me the reasoning based on whatever the answer is for each one because I do not fully understand Countable Infinite and Uncountable Infinite sets

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  • $\begingroup$ Then read up on countable infinite vs. uncountable infinite. There is nothing we can say that can't be said better in a good text. $\endgroup$ – fleablood Dec 19 '18 at 6:37
  • $\begingroup$ @fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given. $\endgroup$ – Viserom Dec 19 '18 at 6:41
  • $\begingroup$ Then it's up to you to ask us a question we can answer for you. We can't read your mind and know why you don't understand it. $\endgroup$ – fleablood Dec 19 '18 at 6:45
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$\aleph_1$ by definition is the first uncountable cardinal number. So $|A| < \aleph_1$ by definition means that $A$ is a countable set. The reals are not countable.

We (should) know that $|\mathbb{R}| = |P(\mathbb{N})| = 2^{\aleph_0}> \aleph_0$. So in particular $2^{\aleph_0} \ge \aleph_1$. The so-called continuum hypothesis (which is independent of ZFC, so "unknown" in your list) states that $2^{\aleph_0} = \aleph_1$.

By Cantor's theorem $|P(\mathbb{R})| = 2^{|\mathbb{R}|}= 2^{2^{\aleph_0}} > 2^{\aleph_0} \ge \aleph_1$.

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