. Let S be an infinite subset N (SCN). Construct a bijection between S and N....
Suppose |N| ≤ |S|, or in other words, S contains a countably infinite subset. Show that there exists a countably infinite subset A ⊂ S and a bijection between S \A and S.
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.) Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
11. (a) Let A be the open interval (1,5), and let B be the interval (0,8). Define a bijection from A to B (b) Let A = (0,00) and let B = [0,00). Define a bijection from A to B. 12. Is it possible to find two infinite sets A and B such that If your answer is yes, then construct an example 13. Is it possible to find a finite set A such that [AAI = 27? 11. (a)...
Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that the number of subsets of S of size k is equal to the number of subsets of size n- k. Be sure to prove that vour mapping is both injective and surjective. b. Prove, by constructing a bijection, that the number of odd-cardinality subsets of S is equal to the number of even-cardinality subsets of S. Be sure to prove that your mapping is...
1. Let A -(a, b) a, b Q,a b. Prove that A is denumerable. (You may cite any results from the text.) 2. Let SeRnE N) and define f:N-+S by n)- n + *. Since, by definition, S-f(N), it follows that f is onto (a) Show that f is one-to-one (b) Is S denumerable? Explain 3. Either prove or disprove each of the following. (You may cite any results from the text or other results from this assignment.) (a) If...
Let (X, d) be an infinite discrete metric space. Prove that any infinite subset of X is closed and bounded but NOT compact
3. Use type vectors to establish the bijection (mentioned in the proof of Theorem 2.4.1) between partitions of n into k parts with smallest part at least 2 and partitions of n-k into k parts. 3. Use type vectors to establish the bijection (mentioned in the proof of Theorem 2.4.1) between partitions of n into k parts with smallest part at least 2 and partitions of n-k into k parts.
18. Prove the infinite pigeonhole principle, that is, let S be an infinite set, n E Zt. Prove that no matter how the elements of S are partitioned into n parts, at least one of the parts must be infinite 18. Prove the infinite pigeonhole principle, that is, let S be an infinite set, n E Zt. Prove that no matter how the elements of S are partitioned into n parts, at least one of the parts must be infinite
mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism. mophisn Define an equivalence relation on Rbyy Z and let /Z be the resulting quoi ant rane. Carefully construct a continuous bijection from R/Z. to the circle S(,y) E R+ 1) and prove that it is a homeomorphism.
Exercise 2.20 Let S= {v, V2, ... , Vpf be a subset of R" containing n vectors. Prove that if S generates R and S is linearly independent, then n m. Exercise 2.20 Let S= {v, V2, ... , Vpf be a subset of R" containing n vectors. Prove that if S generates R and S is linearly independent, then n m.