(a) Use the Quine-McCluskey method to find all prime implicants of $f(a, b, c, d, e)=\Sigma m(1,2,4,5,6,7,9,12,13,15,17,20,22,25,28,30)$. Find all essential prime implicants, and find all minimum sum-of-products expressions. (b) Repeat part (a) for f'.

Toe Computer Security Institute (CSI) conducts an annual survey of computer crin1e at U.S. businesses. CSI sends survey questionnaires to computer security personnel at all U.S. corporations and government agencies. Toe latest CSI survey was sent by post or email to 5,412 firms, and 351 organizations responded. Forty-one percent of the respondents admitted unauthorized use of computer systems at their firms during the year. (CSI Computer Crime and Security Survey, 2010/2011.) a. Identify the population of interest to CSI. b. Identify the data collection method used by CSI. Are there any potential biases in the method used? c. Describe the variable measured in the CSI survey. Is it quantitative or qualitative? d. What inference can be made from the study result?

Let $[G, \cdot]$ be a commutative group with identity i. For a fixed positive integer k, let $B_{k}=\left\{x | x \in G, x^{k}=i\right\}.$ Prove that $\left[B_{k}, \cdot\right]$ is a subgroup of $[G, \cdot].$

(a) Use the are length formula for parametric equations to set up the integral for finding the length of each curve on the given interval. (b) Find the length s of each curve on the given interval. (c) Graph each curve over the given interval. $x(t)=t-\cos t$, $y(t)=1-\sin t$, $0 \leq t \leq \pi$

Write the d’Alembert solution for the problem $u_{t t}=c^{2} u_{x x}$ for $-\infty<x<\infty, t>0$ and $u(x, 0)=f(x), u_{t}(x, 0)=g(x)$ for $-\infty<x<\infty$. $c=1, f(x)=x^{2}, g(x)=-x$

A and B are subsets of a set S. Prove the following set identities by showing set inclusion in each direction.

$$\left.\begin{array}{l} \text { a. }(A \cup B)^{\prime}=A^{\prime} \cap B^{\prime} \\ \text { b. }(A \cap B)^{\prime}=A^{\prime} \cup B^{\prime} \end{array}\right\} \text { De Morgan's laws }$$

c. $A \cup(B \cap A)=A$ d. $\left(A \cap B^{\prime}\right)^{\prime} \cup B=A^{\prime} \cup B$ e. $(A \cap B) \cup\left(A \cap B^{\prime}\right)=A$ f. $[A \cap(B \cup C)]^{\prime}=A^{\prime} \cup\left(B^{\prime} \cap C^{\prime}\right)$

Find the inverse Laplace transform of the function. $\frac{Z^{2}}{(z-2)^{3}}$

(Requires calculus) a. Expand $(1+x)^{n}.$ b. Differentiate both sides of the equation from part (a) with respect to x to obtain $n(1+x)^{n-1}=C(n, 1)+2 C(n, 2) x+3 C(n, 3) x^{2}+\cdots+n C(n, n) x^{n-1}$ c. Prove that $C(n, 1)+2 C(n, 2)+3 C(n, 3)+\cdots+n C(n, n)=n 2^{n-1}$ d. Prove that $C(n, 1)-2 C(n, 2)+3 C(n, 3)-4 C(n, 4)+\cdots+(-1)^{n-1} n C(n, n)=0$

Refer to the Business and Society (Mar. 2011) study on the sustainability behaviors of CPA corporations. Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each 992 senior managers at CPA firms. Numerical measures of both central tendency and variation for level of support are shown in the MINITAB printout below. Give an estimate of the number of the 992 senior managers who have support levels between 41 and 95 points.

$$\begin{matrix} \text{Variable} & \text{N} & \text{Mean} & \text{StDev} & \text{Minimum} & \text{Median} & \text{Maximum}\\ \text{Support} & \text{992} & \text{67.755} & \text{26.871} & \text{0.000} & \text{68.000} & \text{155.000}\\ \end{matrix}$$

Let

$$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$$

and assume det(A)=10. Find

$$\operatorname{det}\left[\begin{array}{lll} a & g & d \\ b & h & e \\ c & i & f \end{array}\right]$$

.

Find the indicated term in the expansion. The fourth term in $(a+b)^{10}$

Use > or < to compare the whole numbers. $74,328 \square 74,238$

Suppose that G is a simple group and $f: G \rightarrow H$ is a surjective homomorphism of groups. Prove that either f is an isomorphism or $H=\langle e\rangle$.

Explain what problem can arise if an item stored in a hash table is later deleted.