# Countable Infinite and Uncountable Infinite sets

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

• 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. – fleablood Dec 19 '18 at 6:37
• @fleablood have read about it and even watched youtube clips, but still don't know how to answer the questions given. – Viserom Dec 19 '18 at 6:41
• 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. – fleablood Dec 19 '18 at 6:45

$$\aleph_1$$ 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 $$|P(\mathbb{R})| = 2^{|\mathbb{R}|}= 2^{2^{\aleph_0}} > 2^{\aleph_0} \ge \aleph_1$$.