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Definition 0.1. Let G (V,E) be graph. (That is. V s a set and EC {(u, of u,ve . For k N+ a A-coloring of Gisa function c:V -> {1,.,If such that for all u, then c(u) c(v). . We define RG V V :u=0v there is path to in G). This an equivalence relation and its equivalence classes are precisely the connectivity class of G. We then define x(G) /VVRc| of connectivity classes of G Problem 1. Let G (V.E) be graph with at lenst one edge. Prove that there is partition {V1,V2} of V such if and only if G has 2-coloring In one direction of your proof, you will have to define the 2-coloring c: V {1,2}, and that step shoukd look like "We choose c(v): {: Definition 0.2. The class of forests, Forests is defined recursively as follows: Base. If is single element then (101.0) in Forests. Allowed step 1. F1 = (V1,E1) and F2 (V2,E2) are in Forests and = D, then is in Forests, where Allowed step 2. IF (V,E) is in Forests, V. and EV. then F (V¹,E') in Forests. where V=V U{x) and E' Nothing else Forests. Problem 2. Use structural induction relative to the recursive definition of Forests above to prove that for every forest F (IV. E), (E)=MV x(F). [Note: There is proof of this fact that avoids induction in favor of previously proved facts (where you used structural induction), but want to see the proof by structural induction here. Grapha are undirected and they are not to haver any loops. That a (V,E), then E 2-element 1-elernert auberta (which reprement loops) are not allowed 1 Lemma 0.3 For everyin N+ for any family of sets. such tha =n, F either S Bor BC then there is set FES suchthatFC A forall Ae Problem 3 Prove that for every N+.for any family of sets F such that .. n. if for all A. B € . either C Bor BC A, then the sets in F can be ordered to form an ascending chain that is, there bijection S Fsuch that S(1) C S(2) S(n). |Hint Proceed by induction or strong induction, and use Lemma 0.3 get back to smaller creses You do not need prove Lemma 0.3 again. Problem 4. Let Vbeanon-emply set, and let f:Vx V-> Vbea function We make few definitions Let SCV. We say that Sis f-closed if for all S. Let x CV. Then define F3 = {SCV:Sis f-closed and xgs). and we define K(X) n°x (This K(X) should be called the top-down -closure of X.) Again, let XCV. Then we define Co(X): X, Cnt(X)= Ca(X)U{f(a,v) € C2(X)} for ench nen and C(X) =U,Ca(X) (This C(X) should be called the bottom-up f-closure of x.) Let XCV a) Let x CV. Show that K(X)is f-closed (It follows that K(X) itselfis in Fx: hence, K(x) the smallest f-closed set containing x. You don't need to prove this.) b) Show that C(X) f-closed. (It follows that K(x)C c(x) because C(X). Again, you don't need to prove this.) c) Using induction or strong induction on n. show that G (XX) S K(X) for every Use this fact toconclude that C(X) K(X) d) Conclude that C(X) K(X). so we em drop the words "top-down" and "bottom-up." Problem 5. Again, let V be non-empty set. and let f VxV-> V be function We make a few more definitions: An f-congruence is an equivalence relation ECV (V such that for all u,r EV, uEd AvEv ==> IfECV V is an (-congruence, then we define a) Let ES V be an f-congruence. Prove that fr is e function from V/E V/E toV/E b) Let W be another non-empty set, and let g W w-> W be a function. Also. let 4 V-> W be a surjective function such that for Showthat Ex is an f-congruence. (We know from class that Exp is an equivalence relation, so you don't need to recoppitulate that part of the argument. |Recall that V/E#=24 wew} the partition of V associated with 4. (You need not prove this.) Then. we have bijection is V/Eg - W 4(4-1(20)) w. In particular, for 21 € V. (Le(w)] and c) With W. and asin part (b). show that for all V. [v])) 3 |The answer here is just calculation using facts already noted above.| interrented mears abso an This 1B,1 the firat thenorem

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