Recall a set A is countable if |N| ≥ |A|. Recall that Cantor’s theorem shows that P(N) = {X ⊆ N} is uncountable. Prove this.Next,show that F = {X ⊆ N | |X| < ∞} is countable, with an onto function g : N → F that is computable.

Q: Let N be the set of all ounting numbers: 1, 2, 3,.... Consider these set definitions: A = {z € N| (4…

A: The complete answer is given below.

Q: Write A greedy algorithm to find the maximal independent sets for matroids step by step.

A: A greedy algorithm to find the maximal independent sets for matroids step by step.

Q: The goal of this question is to give a simple proof that there are decision problems that admit no…

A: Given, Define Σ+ as the set of all binary strings, i.e., Z+ = {0, 1, 00, 01, 10, 11, 000, 001,...}.…

Q: To prove a set A is a subset of a set B, we must: O Show for all x, if x is in A then x is in B O…

A: The solution to the given problem is below.

Q: Sets 1. Let A, B, C C U be subsets of some universe U. Let S = [(AN B) \ C]U[(AnC)\ B]U [(BnC) \ A].…

A: Answer S contains all the elements which are contained in exactly two subsets, i.e. if x is an…

Q: Suppose (S, I) is a matroid, and suppose I' = { A' | S – A' contains some maximal A∈I }. Show that…

A:

Q: Suppose we wanted to show that if Sis a finite set, then |2| = 2SI (where 2 is another way of…

A: Suppose we wanted to show that if S is a finite set, then |251 = 215 (where 25 is another way of…

Q: Create a Turing machine that computes a mapping reduction from L1 ≤ m L2 where L1 ＝ {w : w ∈ {a,b}*…

A: The complete  answer is given below .

Q: s A ⊆ B if  A = {8, 10, 12, 14}  and  B = {8, 10, 12, 16, 20}?

A: Here,  The elements of set A are { 8, 10, 12, 14 } And the elements of set B are { 8, 10, 12, 16, 20…

Q: Consider the Fibonacci sequence F(0)=0, F(1)=1 and F(i)=F(i-1)+F(i-2) for i > 1. For the sake of…

A: Answer is given below .

Q: Let L be a language over the alphabet Σ = {a,b,c}. Prove that L = {abman for any positive integers n…

A: Regular language is a language which can be expressed using a regular expression or using finite…

Q: Let A1, A2,..., An CU be n subsets of some universe U satisfying that A1 Ç A2 Ç A3 Ç .… Ç An. Prove…

A: The handwritten answer is to be solved below:

Q: [Undecidability and Reductions] OVERLAPPING = {{M, D) : | M is a Turing Machine and D is a DFA such…

A: A reduction from ATM to Overlapping is : Input : &lt;M,w&gt; Create a DFA D which accepts only one…

Q: Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. Describe an efficient…

A: Given that, Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. only allowed…

Q: 1. Reword the statement below as a theorem about graphs and then prove it. Assume that if A is a…

A: We use the pegionhole principle to explain this. In  mаthemаtiсs,  the  рigeоnhоle  рrinсiрle…

Q: For E= {a, b}, construct dfa's that accept the sets consists of (i) All strings with no more than…

A: all dfa are given below:-

Q: Show that f(n) is Θ(g(n)) for the correct function g(n). Prove your result using the definitions…

A: f(n) = n 1.5 log(2n) + n 2 log(n 2 ) + √ n:-

Q: Prove that L is undecidable by finding a reduction from  to it, where ={| M is a Turing machine and…

A: L is undecidable by finding a reduction from  to it, where ={&lt;M,w&gt;| M is a Turing machine and…

Q: Let S be the set of all sentences in English Wikipedia as it existed on Feb 7, 2022 at midnight.…

A: S is a language that can be described using deterministic finite-state automaton. S is a language…

Q: Graphs G and H are isomorphic if the nodes of G may be reordered so that it is identical to H. Let…

A: Graphs G and H are isomorphic if the nodes of G may be reordered so that it is identical to H. Let…

Q: Given a set S of integers, we say that S can be partitioned if it can be split into two disjoint…

A: Answer is given below .

Q: Draw out a determinism Turing machine that computes a mapping reduction from L1 ≤ m L2 where L1 ＝…

A: