What Happened 9/7/17
Today we started class by discussing one final topic from Ch 1: Cardinality. Cardinality is a term used to refer to the size of a set. In discussing cardinality, we reviewed what it means for two sets to have a 1-1 correspondence, which can be defined as a function that is both 1-1 and onto. This is important to define as two sets A and B have the same cardinality if there exists a 1-1 correspondence. One example we discussed was the 1-1 correspondence that exists between E={2,4,6,…} the set of all even natural numbers and N the set of natural numbers. Then we discussed that if a set A has the cardinality as the set of natural numbers N, then the set A is countable. If an infinite set is not countable, we call it an uncountable set. Therefore, the set of rationals Q is countable and the set of real numbers R is uncountable. After this, we began to discuss topics from Ch 2: Sequences. A sequence is a function whose domain is the set of natural numbers N. Then we discussed the convergence and divergence of a sequence. In this discussion, we were presented with definitions that were much thorough than the ones we were given in Calc II. We then finished class by learning how to go about proving that a function either diverges or converges.
Good synopsis, Jonathan. There are two points that need further clarification:
1) Regarding finite sets and “countability,” I would caution against the statement “Finite sets are also countable.” IN THIS TEXTBOOK, “countable” is defined as being in 1-1 correspondence with the natural numbers; which is never true for a finite set. IN OTHER TEXTS, “countable” is more general defined as containing AT MOST countably infinitely many points; which is possible for finite sets and countably infinite sets. So, take particular care if you read material from other sources; if they use the word “countable” you need to consider which definition of the word they are using.
2) We will be talking about convergence A LOT in this class, so it is important to pay attention to terminology and details. In this lecture, we ONLY discussed convergence of SEQUENCES of real numbers. We will quickly get into convergence of SERIES and general FUNCTION convergence, but we have not discussed these precise concepts yet in class.