During I.A. on Tuesday (Holy Cow 7:25 am is early to start a class) two students presented problems at the board. I really enjoy having this part of the class. All of these students have taken courses from a colleague of mine. She has made oral presentation of homework a regular part of her upper level courses. So, they are all used to it, but it is fairly new to me. I enjoy hearing the thought processes of the students as they assimilate this knowledge. I even developed Rubric to grade these presentations. Don’t I sound like an education geek? (By the way, I was never formally trained to teach.)
We then covered a few more results regarding countability, such as categorizations of countability, the fact that the countable union of countable sets is countable, and the countability of the rationals. We then diverted from the text a bit and introduced the concept of cardinality and cardinal numbers. I introduced them to aleph-null as the “size” of the natural numbers and the smallest of all transfinite cardinal numbers. We, then, proved that the “size” of the reals is strictly larger than the set of natural numbers, i.e., the Reals are uncountable. I ran out of time before covering Cantor’s Theorem. It is one of my favorite proofs in this class so there’ll be an entry in the next day or two over just that. I’d describe it as “slick.”
Next time, we finally finish Chapter 1 and begin covering The Real Numbers. What the heck are they anyway? We’ll using the axiomatic approach.
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