Question 1 (12 Points): 1. Convert the following decimal numbers to binary representation: a. 12 b. 134

2. Convert the following binary number to decimal: a. 1101111 b. 11001011

Question 2 (8 Points): 1. Prove that the following logic statement is logically true using truth table: ((pr)(qr))((pq)r))

2. Prove that the following logic statement is logically true using proof by contradiction: (pq)(qr)(pr)

Question 3 (12 Points): Let A = if x > 0, then x + 4 > 3, B = (x+y)2 = x2+2xy+y2 , and C = 0 > 1 OR 22 = 4. Determine truth value of the following Boolean expressions:

a. A

b. AB

c. ABC

d. ABC

e. (AB)C

f. (ABC) ((AB)(AC))

Question 4 (12 Points): Simplify the following Boolean expressions using Boolean identities discussed in Module 2:

a. (A+C)*(A*D+A*~D)+A*C+C

b. ~A*(A+B)+(B+A*A*(A+~B))

c. [A*~B*(C+B*D)+~A*~B]*C

d. ~((A+B)+(~A+B)*D)+~A*B*(~C+D)+(A+B*C)*(C+D)

Question 5 (9 Points): Simplify the following circuit by answering each of the steps below.

(1) What is the Boolean expression for the above circuit?

(2) Simplify the Boolean expression from (1).

(3) Draw the circuit from the simplified Boolean expression from (2).

Question 6 (9 Points): Draw the circuits of the following Boolean expressions using OR, AND, and NOT gates: (Both 2-input and 3-input gates are allowed. Also, please don't simplify the Boolean expressions)

a. (A+B)*(B+C)*(C+~A)

b. ~A*~B*C+~A*B*C+A*~B

c. A*~B*C+A*B*C+(C+D)*(~D+E)

Question 7 (9 Points): Simplify the following circuit by answering each of the steps below.

(1) What is the Boolean expression for the above circuit?

(2) Simplify the Boolean expression from (1).

(3) Draw the circuit from the simplified Boolean expression from (2).

Question 8 (8 Points): 1. Convert each of the following twos complement to Decimal

a. 01011100

b. 11010010

2. Convert the following decimal to eight-bit twos complement binary numbers.

a. -127

b. 27

Question 9 (12 Points): Add the following two's complement binary numbers and also express the answer in decimal. Identify each case in which the answer is incorrect because of overflow.

a. 00001111 + 01010011

b. 11111111 + 01101110

c. 10000000 + 11000011

d. 11110000 + 11111000

Question 10 (9 Points): Consider the following addition problems. Please first convert each decimal number into eight-bit twos complement notation, perform the addition in twos complement notation, then convert the results back to decimal numbers. Identify each case in which the answer is incorrect because of overflow. a. 23+45 b. 23-45 c. 128+128

please answer question 3 all then rest question answer

Question 5 (9 Points): Simplify the following circuit by answering each of the steps below. (1) What is the Boolean expression for the above circuit? (2) Simplify the Boolean expression from (1). (3) Draw the circuit from the simplified Boolean expression from (2). Question 7 (9 Points): Simplify the following circuit by answering each of the steps below. A B (1) What is the Boolean expression for the above circuit? (2) Simplify the Boolean expression from (1). (3) Draw the circuit from the simplified Boolean expression from (2)