Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for 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...
. Let S be an infinite subset N (SCN). Construct a bijection between S and N. Proof.
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
(a) Prove that a set Ti is denumeratble if and only if there is a denumerable set T2. bijection from Ti onto a -2- (b) Prove in detail that if S and T are denumerable, then S UT is denumerable. (c) Prove that the collection F(N) of all finite subsets of N is coumtable (a) Prove that a set Ti is denumeratble if and only if there is a denumerable set T2. bijection from Ti onto a -2- (b) Prove...
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
. Let C be a collection of open subsets of R. Thus, C is a set whose elements are open subsets of R. Note that C need not be finite, or even countable. (a) Prove that the union U S is also an open subset of R. SEC (b) Assuming C is finite, prove that the intersection n S is an open subset of R. SEC (c) Give an example where C is infinite and n S is not open....
Use induction to prove that every set of n elements has 2n distinct subsets, for all n ? 0. Hint for the inductive case: fix some element of the set and consider whether it belongs to the subset or not. In either case, reduce to the inductive hypothesis.
4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.) 4. Let n be a positive integer. Z" is the set of all lists of length n whose entries are in Z. Prove that Z" is countable. (Hint: Find a bijection between Z"-1x Z and Z" and then use induction.)
Let A and B be subsets of S. Prove the following: 1. The compliment of A is a subset of B iff A union B = S 2. A is a subset of the compliment of B iff B is a subset of the compliment of A
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...