This blog will be on the second half of 10.5 - the Schroder-Bernstein Theorem.
This is a very short section. We will talk about the Axiom of Choice.
Theorem: The sets P(N) and R are numerically equivalent.
Corollary: The sets 2^N and R are numerically equivalent.
What was difficult for me?
This section was not difficult for me. It was rather simple after the discussion in class yesterday. The Axiom of choice states For every collection of pairwise disjoint nonempty sets, there exists at least one set that contains exactly one element of each of these nonempty sets.
I enjoy this section, and it is not too difficult for me to understand the theorems. The only thing that is difficult for me is the proving of the theorems. And since this is a proofs class, I need to focus more on this. I enjoy knowing the proofs and seeing how they work - it is the thinking through them in the moment that is difficult for me. I look forward to learning more about how to prove the theorem and corollary above.
What did I enjoy most?
I enjoy counting the cardinality of sets. While at time it is counterintuitive (an infinity can be bigger than another infinity, but they are still numerically equivalent), it does make a lot of sense and is quite fun to work with. I enjoy showing that there are one-to-one functions and the difference in numerical equivalence. I find this interesting.
This is a very short section. We will talk about the Axiom of Choice.
Theorem: The sets P(N) and R are numerically equivalent.
Corollary: The sets 2^N and R are numerically equivalent.
What was difficult for me?
This section was not difficult for me. It was rather simple after the discussion in class yesterday. The Axiom of choice states For every collection of pairwise disjoint nonempty sets, there exists at least one set that contains exactly one element of each of these nonempty sets.
I enjoy this section, and it is not too difficult for me to understand the theorems. The only thing that is difficult for me is the proving of the theorems. And since this is a proofs class, I need to focus more on this. I enjoy knowing the proofs and seeing how they work - it is the thinking through them in the moment that is difficult for me. I look forward to learning more about how to prove the theorem and corollary above.
What did I enjoy most?
I enjoy counting the cardinality of sets. While at time it is counterintuitive (an infinity can be bigger than another infinity, but they are still numerically equivalent), it does make a lot of sense and is quite fun to work with. I enjoy showing that there are one-to-one functions and the difference in numerical equivalence. I find this interesting.
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