#Cardinality proofs

12 messages · Page 1 of 1 (latest)

serene hare
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Hello, I need help with these 5 proofs

1.) |(0,1) x (0,1)| = |(0,1)|
2.) |R x R| = |R|
3.) Prove that {0,1}xR is uncountable
4.) and 5.) are in the image, I am to prove the countability or non countability of the sets (I think 4 is countable and 5 is not but i cant prove it)

For 1 and 2, the directions specifiy that I have to construct a bijection between both sets, so no injecting both ways or using another set to act as a "middle point" for the equality.

I have access to the fact that the carteisian product and union of countable sets is countable, and the countability/non-countability of R, Q, N, etc...

Please help me, This problem set has taken me over 35 latex pages to answer and these are the last 5 problems. My professor and TAs do not respond to emails and there is no support system set up for my class, I am in a panic right now. These proofs seem unreasonably hard. For context, this is an introduction to mathematical proofs course.

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surreal spruceBOT
true stag
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First of all, do you see why $(0,1)$ has the same cardinality as $\mathbb{R}$?

robust aspenBOT
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bondalton

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serene hare
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Yes

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bc there is a bijection between them

true stag
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So you can agree that 2) follows from 1) right?

serene hare
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For 1 and 2, the directions specifiy that I have to construct a bijection between both sets, so no injecting both ways or using another set to act as a "middle point" for the equality.

true stag
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Ohhh, okok

serene hare
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for |RxR| = |R|, I tried to interleave the decimal expansions in the two reals in the pair from RxR into a single real number, but i cant figure out how to deal with for example 0.50000... and 0.499999.... being the same number

true stag
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I found this really good solution dealing with that problem, which in turn can be fixed