7. Let G be a loop-free undirected graph, where ∆ = maxv∈V {deg(v)}.a) Prove that χ(G) ≤ ∆ + 1.b) Find two types of graphs G, where χ(G) = ∆ + 1.

3. Consider the map Xn+1 = aX, (1 – X;), where a > 1. (i) Find the fixed points of the map. (ii) Find the range of values of a for which the nontrivial fixed point is linearly stable.

3. Let A = {1,2,3), let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) E R (XnY=0) ^ (XUY = A). (a) Prove or find a counterexample for the following properties: • antisymmetry. asymmetry. symmetry. irreflexivity/ • reflexivity. • transitivity • uniqueness

3. Let A = {1, 2, 3}, let S = P(A) and let RC S x S be a relation defined by the rule (X,Y) ER⇒ (XnY=0)A (XUY = A). (a) Prove or find a counterexample for the following properties: · antisymmetry. asymmetry. • symmetry. ● irreflexivity/ • reflexivity. • transitivity • uniqueness (b) Based on the findings from the previous point, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.

1. (a) Let r(t) =. Sketch the path r(t) for t between -2 and 2, showing the direction of increasing t. Make sure you give the coordinates of at least 3 points that are on this path and label your axes.

-by. Suppose f(x, y) = xy - ax (A) How many local minimum points does ƒ have in R²? (The answer is an integer). (B) How many local maximum points does f have in R²? (C) How many saddle points does f have in R²?

1 Let f(r) = (r – 1)², g(x) = 1– a², h(x) = 2(r + 3)2 – 8, find for each function a) r- and y- intercepts. b) vertex c) graph

1. Let C be the closed loop that is traversed from (0,0) to (2,4) along the line y and then returns from (2, 4) to (0,0) along the parabola y evaluate fo(xy + e3")dx – x² dy. Include a sketch and show all work. 2x x2. Use Green's Theorem to -

Please write it in detail.

Let P(Z) = 0.42, P(Y) = 0.39, and P(ZnY) = 0.16. Use a Venn diagram to find (a) P(Z'nY'), (b) P (Z'UY'), (c) P(Z'UY), and (d) P (ZnY'). (a) P(Z'nY') (b) P (Z'UY') (c) P (Z'UY) = (d) P (ZnY') = = = (Type an integer or a decimal.) (Type an integer or a decimal.) (Type an integer or a decimal.) (Type an integer or a decimal.) .. Z Y CO