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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->

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