3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from class. (You need to prove that your function is a bijection.)
3. (8 marks) Let be the set of integers that are not divisible by 3. Prove that is a countable set by finding a bijection between the set and the set of integers , which we know is countable from cl...
Let be the set of odd integers. Let . a) Determine a bijection from to . b) Is ? Explain. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
7. Prove or disprove: If we know that 2X +6=4 (mod 8), then X +3 = 2 (mod 8). 8. Prove or disprove: If we know that 2X+6 = 4 (mod 7), then X+3 = 2 (mod 7). 9. Let S be the set {311, 254, -172,45,2019, 111,3}. Find a subset T such that the sum of the elements in divisible by 7
We were unable to transcribe this imageWe were unable to transcribe this imagec) Let y - Ara. Suppose that we see when p - 2 and w - 1 then x-3 and y-8. Also when p-1 and w 1.5, then x-4 and y - 10. Can we identify the parameters of the production function from these two observations? Graph what an economist who didn't know the functional form of the production function would conclude about the production set. c) Let...
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space to itself, and is an eigenvalue of , then is a simple eigenvalue of if . 1. Suppose is a vector space of dimension over field where you may assume that is either or , and let be a linear map from to . Show...
Problem 8. Given each pair of sets, come up with a formula for a bijection between them You do not need to prove your function is a bijection. Your formula should not be complicated by any means 1. From (0, 1) to (211, 2019) 2. From [0, 1) to (0, 1] 3. From NU (o) to N. 4. From the set of even numbers to 2 5. From the set of odd numbers to Z. 6. r2'2 7. From R...
Question 8, please. 2. Prove: (a) the set of even numbers is countable. (b i=1 3. The binary relation on pair integers - given by (a,b) - (c,d) iff a.d=cbis an equivalence relation. 4. Given a graph G = (V, E) and two vertices s,t EV, give the algorithm from class to determine a path from s to t in G if it exists. 5. (a) Draw a DFA for the language: ( w w has 010 as a substring)....
You may assume there are exactly simple graphs with vertex set We were unable to transcribe this image1, 2, V3, ..., Unf D)Explain why there are exactly 2) simple grahs with vertex sein which every vertex has even degree. (Hint: Establish a bijection between from this set to the set of all graphs with vertex set {vi, , Vn-1)). 2) Prove that the probability that a randomly chosen simple graph with vertex set {vi,... . vn) wil have an Eulerian...
From the class Introduction to Abstract Algebra on the section of countable and uncountable sets 3. Let X and Y be two nonempty finite sets. Let F(X, Y) denote the set of all function from X to Y. Is this set finite, countably infinite, or uncountable? Prove your answer
5. (FP1.73) Let a and b be positive integers such that a2b and a is even. Then 8 divides a We will prove this in steps. (a) Come up with an example of positive integers a and b such that a2 (b) Now prove the statement. You may use the work from class, but your proof should be and a is even written out as paragraphs and displayed math that flows. It should not be numbered substeps. In other words,...