Honours Analysis Math Problem Example | Topics and Well Written Essays - 750 words - 1

Honours Analysis - Math Problem ExampleCharacterizing of the Cantor set in legal injury of ternary is done when a real number from the closed real musical interval 0, 1 belongs to Cantor set with a ternary expansion containing digits 0 and 2. To relieve oneself this expansion, one has to consider the points in closed real interval 0, 1 in terms of base 3 notation.When constructing the Cantor Middle Third set, we start with the interval 0, 1 removing the middle thirds , this leaves 0,1/3 2/3,1.The next step is to besides remove the middle thirds (1/9, 2/9) and (7/9, 8/9) from the remaining two intervals. This physical process is repeated continuously. From the results, we can note that on the whole the endpoints remain, which are the Cantor set.The total length of the intervals removed in the construction of the Cantor set can be determined as follows. From interval 0, 1 we first remove a middle third interval 1/3 second step we remove two middle intervals of 1/9. We continue wi th the process so that at the nth stage we remove 2n-1 intervals with the length 3-n. The total sum of the removed intervals isIt can be proven that the Cantor set is perfect tense and totally disconnected. In this case, x and y are two distinct points in the cantor set. Since x y therefore x - y. As we can see there is a natural number N that exists in the interval. Next we identify that Cantor set Ck for all k, such that x, y Ck. For each 2N disjoint closed interval from CN there is. Therefore, x and y are inside distinct closed intervals in CN. The two intervals should have an kick in interval between them, which is not part of the Cantor set otherwise this would be a single closed interval. The chosen point can be represented by z, therefore z Cantor set and it is between x and y. (Gordon, 1994, p. 301)If we put f in its inverse If x Q, then also - x Q. therefore f o f (x) = f ( f (x)) = f ( - x) = - (-x) = x. If x Q then f o f (x) = f (f (x)) = f (x) = x. Thus for all x R, we have that f o f