complwx analysis 7. Show that the accumulation points of any set form a closed set.
1. Prove that for any set S S R, S is closed if and only if Se is open. Notice the book has a proof of this, but it uses a different notation for set complements and a different definition of neighborhood. You may consult it, but you must write your proof using the definition for interior point I presented in class (also in the notes on blackboard). If you copy the proof from the book you will not receive...
8. Show: An infinite set of points need not be a closed set.
1. Find the boundary and the interior for the following sets. Find the set of all accumulation points and the closure for the following sets. Classify each set as open, closed, or neither closed nor open. Use Heine-Borel theorem to determine whether it is a compact subset of R. A is closed/ open / neither closed nor open A is compact /not compact intB B is closed / open / neither closed nor open B is compact / not compact...
Prove that a bounded set in R2 with a finite number of accumulation points has content 0.
Prove that a bounded set in R2 with a finite number of accumulation points has content 0.
Analysis Show that the least upper bound of any nonempty set of real numbers is unique
I need 7 - 10. Ignore others please!
1. (10 points) True/False. Briefly justify your answer for each statement. 1) Any subset of a decidable set is decidable 2) Any subset of a regular language is decidable 3) Any regular language is decidable 4) Any decidable set is context-free 5) There is a recognizable but not decidable language 6) Recognizable sets are closed under complement. 7) Decidable sets are closed under complement. 8) Recognizable sets are closed under union 9)...
b. Consider the set of numbers T = {1,4, 7, 10, 13, , 3n + 1,...]. Show that T is closed under multiplication. Call p e T a mock-prime if the only factors of p in T are 1 and p, p> 1. Show that every number in T can be factored as a product of mock-primes, but not necessarily uniquely.
b. Consider the set of numbers T = {1,4, 7, 10, 13, , 3n + 1,...]. Show that T...
7. (5 points) Show that the following set is infinite by placing into a one-to-one correspondence with a proper subset of itself. Show the general terms of the set and subset. (a) [3,6,9,12,15...
2. Consider the set of all polynomials of the form 2 + at + bt2 where a and b are real number (a). Show by means of an example that this set is not closed under polynomial addition. (b) Show by means of an example that this set is not closed under scalar product.