Prove for the following:
a. Theorem: (Cantor-Schroder-Bernstein in the 1800s) For any set
S, |S| < |P(S)|.
b. Proposition N×N is countable.
c. Theorem: (Cantor 1873) Q is countable. (Hint: Similar. Prove
for positive rationals first. Then just a union.)
1. Prove that the Cantor set contains no intervals.
2. Prove: If x is an element of the Cantor set, then there is a
sequence Xn of elements from the Cantor set converging
to x.
Prove that the function defined to be 1 on the Cantor
set and 0 on the complement of the Cantor set is discontinuous at
each point of the Cantor set and continuous at every point of the
complement of the Cantor set.
1.) Prove that Z+, the set of positive integers, can be
expressed as a countably infinite union of disjoint countably
infinite sets.
2.) Let A and B be two sets. Suppose that A and B are both
countably infinite sets. Prove that there is a one-to-one
correspondence between A and B.
Please show all steps. Thank you!
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