Question

Please show all the work!!! Thank you

1. The Cantor set is one of the most famous sets in mathematics and has some rather unique properties. The Cantor set was discovered in 1874 by Henry John Stephen Smith and introduced to the world by George Cantor in 1883. The Cantor set is a set of points lying on a single closed line segment, say from [0,1]. It is constructed as follows: Start with the closed interval Co-10.1]. Remove the open interval G. This leaves two closed intervals, C1-0 and [h]. Next, remove the middle third from each of these intervals, giving you 4 intervals now. Continue to remove the middle third interval from each of the previous intervals. The Cantor set Co is what remains at infinity; that is, when you've removed the middle third from the intervals infinitely many times. a. b. c. d. e. Draw the first several stages of the Cantor set, say Co to Ca How many intervals are there at each stage Co to C1o Come up with a formula for the amount of the interval removed at the nth stage. Find the total amount of the interval removed at C1o Find the total amount of the interval removed, that is, find the sum of the infinite series. How much of the initial interval remains? f. g. h. Find at least 5 numbers that are in the Cantor set. Is the value ¼ in the Cantor Set, Ca, In fact, the Cantor set contains infinitely many numbers. You start with an interval of length 1. You remove a total length of 1. Yet, the remaining interval contains an infinite number of points. Cantor consider this a perfect set that is nowhere dense.

Answer 1

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Similaty 2-19 To @ in the nth Sfegethe amuh