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I missed a lecture for my set theory class, so I'm stuck on the following homework problems:

Prove that each of the following sets has the power of the continuum:

a. The set of all infinite sequences of positive integers; b. The set of all ordered n-tuples of real numbers; c. the set of all infinite sequences of real numbers;

From google I was able to find out the definition of the power of continuum but as far as any equivalence proofs I'm stumped. Any insight would be appreciated!

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    Often the easiest way to prove two sets have the same cardinality is to find an injection (1 to 1 map) from one set to the other, and vice versa. Also, if you are willing to believe that no set has cardinality larger than $R$ then only one direction of this is needed for your questions(you would just find an injection from $R$ to the set you have listed in each of your three questions).2011-05-05
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    The question is what are you allowed to use. Have you already covered cardinal numbers and some rules of cardinal arithmetic in class? Do you already know that $|\mathcal P(\mathbb N)|=|\mathbb R|$? Have you already learn Cantor-Bernstein theorem (which is implicitly mentioned in Joe's comment), that $|A|\le|B|$ and $|B|\le|A|$ implies $|A|=|B|$?2011-05-05
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    If we assume as known that the cardinality of the continuum is equal to the cardinality of the set of infinite sequences of $0$s and $1$s, all the rest can be done by fairly simply discovered explicit bijections.2011-05-05
  • 0
    Get the notes from a classmate.2011-05-05

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