Questions 1 and 2 refer to this graph: Note: This is the same graph from the Chapter 11 homework assignment, but on the final exam we are considering vertex E to be the root of the spanning tree. 1 (? points) Using alphabetical ordering with vertex E as the root, draw a spanning tree for this graph by using a depthfirst search. 2 (7" points)Using alphabetical ordering with vertex E as the root, draw a spanning tree for this graph by using a breadthrst search. 3 - (4 points) There are thirty students in a class. Each earned either an A, B, C, D or F on the final exam. How many final exams must I grade to ensure that at least three students have received the same grade? 4 - (6 points) Draw a logic gate diagram for F(x, y)=(xy)+(xy) 5 - (8 points) Determine whether the following statement is True or False using a Truth Table. Assume that x, y, and z represent Boolean variables. (x+y)z=(x+y)z X V Z X (x+y) (x+ y)z x+y ( x+y) ( x+ y) z 1 1 1 0 1 0 1 0 0 0 0 0 0 O 0 0 Are the expressions equal?6 - (6 points) Write a truth table for the proposition: -(paq)v(-p+-r) . Consider a "1" to be true and a "0" to be false. P q r PAq -(qAr -p -r -pa-r -(paq)v(-p+-r) 1 1 1 7 1 1 1 0 0 0 0 1 0 0 0 7 - (6 points) Determine whether (p+q)Ar is equivalent to p+(qAr) . To get full credit on this question, you must show your work AND give an answer stating that the tables are equivalent or not equivalent. Consider a "1" to be true and a "0" to be false. P r p+q (p +q) Ar (qAr) p-(qAr) 0 1 0 O 0 O O 0 O 0 0 0 0 Are the two statements equivalent?8 - (4 points) Suppose that Q(x) is the statement "2x + 2 = x-." What are the truth values of the following statements? Assume x is representing all real numbers. 3xQ(x) (x)OxA 9 -(4 points) Let f and g be functions from the set of real numbers to the set of real numbers defined by: f(x) = 2x- + 2 and g(x) = 2x - 2. Find f - g Find g=f 10 - (4 points) Consider the binomial theorem to expand (2x + y)*. What is the coefficient of the xty? term? You must illustrate use of the binomial theorem for full credit