QuestionQuestion

Transcribed TextTranscribed Text

1. For a set X, the identity function is ix : X -> X such that ix(x) = I for all x E X. (i) Give an example of a function f { a,b} {a, b} such that f#i{a,b} and f is bijective. [1 mark] (ii) Give an example of a function g : {a,b,c} {a,b,c} such that g # i{a,b,c} and gog is bijective. marks] (iii) Suppose h : X X is a function such that hoh is 1-1 (injective). Prove that h is 1-1. 2 marks (iv) Suppose h : X X is a function such that hoh is onto (surjective). Prove that h is onto. [2 marks] 2. the equivalence relation 22 on R defined by The equivalence classes are In = {xER/n<x<n+1}. where="" n="" ez.<br="">(i) Let k E Z. Explain why, if k E In then k =n. [2 marks] (ii) Let = {In /neZ} Prove that I is countable. marks (iii) It is known that R is uncountable. Prove that In is uncountable for every n E Z. [3 marks] </x<n+1}.>3. Let G be an undirected graph, and G = (V,E). Recall that a subgraph of the form {Xp-1,xp}}) is called a path between X1 and Xp in G. and this path has length p - 1. Now, for integer k>1, define Ek = {{x,y} x y and there is a path of length k in G between x and y}, and let Gk = (V,Ek). Note that E1 = E and G1 = G. Thus, for the undirected graph in Figure 1, a b we have {e,b} E E1, {e, d} E E2, {e,d} E E3. etc. C e ( V, E) Figure 1 (i) List the elements of V and E for Figure 1. 2 marks] (ii) What is the length of the longest path in Figure 1? [1 mark] (iii) Draw all spanning trees for the graph in Figure 1. [3 marks (iv) Draw G2, G3, G4 and G5 for the graph in Figure 1. [4 marks] (v) Identify all (if any) cyclic graphs in (iv). [1 mark] (vi) Identify all (if any) connected graphs in (iv). [1 mark] (vii) Among G2, G3, G4 and G5, which (if any) are trees? [1 mark] (viii) In (iv), how many connected components does G4 have? [1 mark] (ix) For this part, consider any G (not just the one in Figure 1). Prove that G is connected if and only if {x,y} E U Ek for every x,y E V such that X # y. [2 marks] (x) Determine the number of graphs (with the same V) that are isomorphic to the graph in Figure 1. [5 marks] 4. Let T be a rooted binary tree of height h. For h>1, we call T a strand if and only if the following holds: (I) there is exactly one leaf and one parent at every level l, for <e<h- -="" 1="" and<br="">(II) there are exactly two leaves at level h. Figure 2 below illustrates three strands T1, T2 and T3. a a a b C b C b C d e e d d f g g f g T1 T2 T3 Figure 2 (i) Is T1 = T2? Is T2 = T3? Justify your answers. [2 marks] (ii) Prove that, for any h > 1, a strand of height h has 2h + 1 nodes. [2 marks] Let N(0) = 1 and, for h>1, let N(h) be the number of different strands of height h. whose nodes are {V1, ...,U2h+1} (iii) Prove that N(h) = 2(2h + 1)hN(h - 1) for every positive integer h. [5 marks] (iv) Use induction to prove that N(h) = (2h+1)! for every positive integer h. [5 marks] </e<h->

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.zip.

    $60.00
    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Discrete Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats