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1.(10%) Given Boolean variables p, q, and r where -p is the complement of p, A is logical AND, v is logical OR, → is logical imply, and is equivalence, prove that the following formula is true.

2.(15%) Given a directed graph G as in Figure 1, derive a relation matrix R of the graph, and derive the matrix of transitive closure R* of relation R.  

3.(15%) In a 2D n ✖ n checkerboard as shown in Figure 2, given two integers p and a (1≤p ≤ q≤n), write the formula of the number of all possible squares in the checkerboard with side length s satisfying p ≤s≤g. Write the result in a formula of n, p, and q.

5.(10%) A complete graph Kn is a graph G(V,E) with n vertices (n ≥ 1) that have an edge between each pair of vertices in the vertex set. Write the formula of the number of all complete suberaphs K, existing in Kn. (1 ≤p ≤n) The formula is a function of n.

6.(10%) Write an algorithm queue partitioning(g. k) to reorder all data elements in a queue q. Assume that the queue q has n integer elements initially. Given an integer k ＞ 1, the reordering will form k ordered partitions of these integers. The resulting queue should be in the order of partition 0, partition 1, and partition k-1. Thepartition of these integer elements contains all integer elements e in the original queue such that e mod k= i. (In this algorithm, you can utilize an empty queue q2 as a local variable.) You can use a function length(q) to get the length of the queue q.For illustration, Figure 3 shows an example with an initial queue state, k = 3, ordered partitions, and a final queue state.

7.(15%) A binary expression tree data structure uses the node data element given as follows. A node either represents a value or represents an arithmetic operator in { +, -, *, / }. An evaluation of an example expression tree is shown in Figure 4. Write a recursive algorithm evaluate(roos) to perform such expression evaluation task where root points to the root element of the expression tree. 

8. (10%) In an undirected graph G(V, E) with n vertices, these is an integer data di in each vertex v. Given a starting vertex S in the vertex set V, write an algorithm that computes the sum of associated data di's of all reachable vertices in V from S. (i.e. A reachable vertex T from the vertex S is a vertex that has a path from S to T.)

4.(15%) Given a bag having 4 white balls and 3 red balls, let us proceed a drawing run that a drawingtakes a ball from the bag, record its color, put it back, and do such drawing for 5 times. Assume thatthe probability of each distinct red ball being drawn is two times of that of a distinct white ball being drawn. Calculate the probability to draw exactly 3 white balls and 2 red balls in the run.

1. An unknown semiconductor has bandgap. of 1.36 eV. It is doped with donors. The effective density of states in the conduction band and the valance band are the same. The Fermi level is 0.2 eV below the bottom of the conduction band. Assume that 30% of total electrons are still in the donor states. Find the intrinsic carrier density in the material at T = 300 K. (20%)

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