Write VHDL code to implement the following logic functions using a 16 words $\times$ 3 bits ROM. W= A′B′C + C′D + ACD′. X = A′C′ + B′D. Y = BD′ + B′C′D

If G is a finite abelian p-group such that $p G=\langle 0\rangle$, prove that

$$G \cong \mathbb{Z}_{p} \oplus \cdots \oplus \mathbb{Z}_{p}$$

for some finite number of copies of $\mathbb{Z}_{p}$.

(a) Implement the following VHDL conditional statement using two 2-to-1 MUXes: F <= A when D = ‘1’ else B when E = ‘1’ else C; (b) Implement the same statement using gates.

Evaluate the determinant of the matrix by inspection.

$$\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & -1 & 0 & 0 \\ 4 & 2 & 2 & 0 \\ 1 & 1 & 6 & 5 \end{array}\right]$$

Determine the values. $_{n} P_{1}$

A gas containing $A\left(2 \mathrm{mol} / \mathrm{m}^{3}\right)$ is fed $(1 \mathrm{m}^{3} / \mathrm{hr})$ to a plug flow reactor with recycle loop $(0.02 \mathrm{m}^{3} \text { loop volume, } 3 \mathrm{kg}$ of catalyst), and the output composition from the reactor system is measured $\left(0.5 \mathrm{mol} \mathrm{A} / \mathrm{m}^{3}\right).$ Find the rate equation for the decomposition of A for the following cases. Be sure to give the units of $-r_{\mathrm{A}}^{\prime}, C_{\mathrm{A}},$ and $k^{\prime}$ in your final expression. Very large recycle, $A \rightarrow 3 R, n=1,50$ A-50% inerts in feed

As a female, how likely are you to follow tradition and change your last name to your husband's when you get married? This was one of the questions investigated in research published in Advances in Applied Sociology (Nov. 2013). According to the study, the probability that an American female will change her last name upon marriage is .9. Consider a sample of 25 American females. How many of these females would you expect to change their last name upon marriage? How likely is it that 20 or more of these females will change their last name upon marriage?

For each of the following binary relations on $\mathbb{R},$ draw a figure to show the region of the plane it describes. a. $x \rho y \leftrightarrow y \leq 2$ b. $x \rho y \leftrightarrow x=y-1$ c. $x \rho y \leftrightarrow x^{2}+y^{2} \leq 25$ d. $x \rho y \leftrightarrow x \geq y$

Refer to the Journal of Systems Sciences & Systems Engineering (Mar. 2013) study of the reliability of a manufacturing system that involves multiple production lines. Consider, again, the network for producing integrated circuit (IC) cards with two production lines set up in sequence. a. Find the mean maximum capacity for each line. Interpret the results practically. b. Find the standard deviation of the maximum capacity for each line. Interpret the results practically.

Shown below is the prime implicant chart for a completely specified four-variable combinational logic function r(w, x, y, z). (a) Algebraically express r as a product of maxterms. (b) Give algebraic expressions for the prime implicants labeled A, C, and D in the table. (c) Find all minimal sum-of-product expressions for r. You do not have to give algebraic expressions; instead just list the prime implicants (A, B, C, etc.) required in the sum(s).

$$\begin{array}{c|cccccccccccc} & 0 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 13 & 14 & 15 \\ \hline \mathrm{A} & \times & \times & & & & & & & & & & \\ \mathrm{B} & & & \times & & \times & & & & & \times & & \times \\ \mathrm{C} & & & & \times & \times & & & & & & \times & \times \\ \mathrm{D} & & & & & & \times & \times & \times & \times & & & \\ \mathrm{E} & & & & & & & & \times & \times & & \times & \times \\ \mathrm{F} & & & & & & & \times & & \times & \times & & \times \\ \mathrm{G} & \times & & & & & \times & & & & & & \\ \mathrm{H} & & \times & \times & \times & \times & & & & & & & \\ & & & & & & & & & \end{array}$$

Which of the following instructions should be privileged? a. Set value of timer. b. Read the clock. c. Clear memory. d. Issue a trap instruction. e. Turn off interrupts. f. Modify entries in device-status table. g. Switch from user to kernel mode. h. Access I/O device.

Prove that the statements are true for every positive integer. $10^{n}+3 \cdot 4^{n+2}+5$ is divisible by 9.

Use the chart above. Write the place of each digit. 4

How many people must be in a group to guarantee that 2 people in the group have the same birthday (don’t forget leap year)?