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Exercises

Hints and Answer Construct a truth table for the following:
1. Hints and Answer
2. Hints and Answer
Construct a truth table for the following:
Hints and Answer Using DeMorgan's Law, write an expression for the complement of F if F(x,y,z) = .
Using DeMorgan's Law, write an expression for the complement of F if F(x,y,z) = xy + .
Hints and Answer Using DeMorgan's Law, write an expression for the complement of F if F(w,x,y,z) = .
Use the Boolean identities to prove the following:
1. The absorption laws
2. DeMorgan's laws
Hints and Answer Is the following distributive law valid or invalid? Prove your answer.

x XOR (y AND z) = (x XOR y) AND (x XOR z)
Show that
1. Using truth tables
2. Using Boolean identities
Show that
1. Using truth tables
2. Hints and Answer Using Boolean identities
Simplify the following functional expressions using Boolean algebra and its identities. List the identity used at each step.
Simplify the following functional expressions using Boolean algebra and its identities. List the identity used at each step.
1. Hints and Answer
2. Hints and Answer
3. Hints and Answer
Simplify the following functional expressions using Boolean algebra and its identities. List the identity used at each step.
1. (ab + c + df)ef
2. x + xy
Simplify the following functional expressions using Boolean algebra and its identities. List the identity used at each step.
1. Hints and Answer xy + x
2. yz + xz
3. wx + w(xy + y)
Use any method to prove the following either true or false:
Hints and Answer Using the basic identities of Boolean algebra, show that:

x( + y) = xy
Asterisk Mark Using the basic identities of Boolean algebra, show that:

x + y = x + y
Hints and Answer Using the basic identities of Boolean algebra, show that:

xy + z + yz = xy + z
Hints and Answer The truth table for a Boolean expression is shown below. Write the Boolean expression in sum-of-products form.

x

y

z

F

0

0

0

1

0

0

1

0

0

1

0

0

0

1

1

1

1

0

0

0

1

0

1

0

1

1

0

1

1

1

1

0
The truth table for a Boolean expression is shown below. Write the Boolean expression in sum-of-products form.

x

y

z

F

0

0

0

1

0

0

1

0

0

1

0

0

0

1

1

1

1

0

0

0

1

0

1

0

1

1

0

1

1

1

1

0
Draw the truth table and rewrite the expression below as the complemented sum of two products:
Given the Boolean function
1. Hints and Answer Derive an algebraic expression for the complement of F. Express in sum-of-products form.
2. Show that .
3. Show that .
Given the function
1. List the truth table for F.
2. Draw the logic diagram using the original Boolean expression.
3. Simplify the expression using Boolean algebra and identities.
4. List the truth table for your answer in Part c.
5. Draw the logic diagram for the simplified expression in Part c.
Construct the XOR operator using only AND, OR, and NOT gates.
Asterisk Mark Construct the XOR operator using only NAND gates.
Design a circuit with three inputs (x,y, and z) representing the bits in a binary number, and three outputs (a,b, and c) also representing bits in a binary number. When the input is 0, 1, 2, or 3, the binary output should be one less than the input. When the binary input is 4, 5, 6, or 7, the binary output should be one greater than the input. Show your truth table, all computations for simplification, and the final circuit.
Draw the combinational circuit that directly implements the following Boolean expression:
Hints and Answer Draw the combinational circuit that directly implements the following Boolean expression:
Find the truth table that describes the following circuit:
Hints and Answer Find the truth table that describes the following circuit:
Find the truth table that describes the following circuit:
Draw circuits to implement the parity generator and parity checker shown in Tables 3.11 and 3.12, respectively.
Draw a half-adder using only NAND gates.
Draw a full-adder using only NAND gates.
Tyrone Shoelaces has invested a huge amount of money into the stock market and doesn't trust just anyone to give him buying and selling information. Before he will buy a certain stock, he must get input from three sources. His first source is Pain Webster, a famous stock broker. His second source is Meg A. Cash, a self-made millionaire in the stock market, and his third source is Madame LaZora, a world-famous psychic. After several months of receiving advice from all three, he has come to the following conclusions:
1. Buy if Pain and Meg both say yes and the psychic says no.
2. Buy if the psychic says yes.
3. Don't buy otherwise.
Construct a truth table and find the minimized Boolean function to implement the logic telling Tyrone when to buy.
Hints and Answer Asterish Mark A very small company has hired you to install a security system. The brand of system that you install is priced by the number of bits encoded on the proximity cards that allow access to certain locations in a facility. Of course, this small company wants to use the fewest bits possible (spending the least amount of money as possible) yet have all of their security needs met. The first thing you need to do is determine how many bits each card requires. Next, you have to program card readers in each secured location so that they respond appropriately to a scanned card.

This company has four types of employees and five areas that they wish to restrict to certain employees. The employees and their restrictions are as follows:
1. The Big Boss needs access to the executive lounge and the executive washroom.
2. The Big Boss's secretary needs access to the supply closet, employee lounge, and executive lounge.
3. Computer room employees need access to the server room and the employee lounge.
4. The janitor needs access to all areas in the workplace.
Determine how each class of employee will be encoded on the cards and construct logic diagrams for the card readers in each of the five restricted areas.
How many 256 x 8 RAM chips are needed to provide a memory capacity of 4096 bytes?
1. How many bits will each memory address contain?
2. How many address lines must go to each chip?
3. How many lines must be decoded for the chip select inputs? Specify the size of the decoder.
Hints and Answer Asterish Mark Investigate the operation of the following circuit. Assume an initial state of 0000. Trace the outputs (the Qs) as the clock ticks and determine the purpose of the circuit. You must show the trace to complete your answer.
Describe how each of the following circuits works and indicate typical inputs and outputs. Also provide a carefully labeled black box diagram for each.
1. coder
2. ltiplexer
Hints and Answer Complete the truth table for the following sequential circuit:
Complete the truth table for the following sequential circuit:
Hints and Answer Complete the truth table for the following sequential circuit:
A sequential circuit has one flip-flop; two inputs, X and Y; and one output, S. It consists of a full-adder circuit connected to a D flip-flop, as shown below. Fill in the characteristic table for this sequential circuit by completing the Next State and Output columns.
Hints and Answer Asterish Mark A Mux-Not flip-flop (MN flip-flop) behaves as follows: If M = 1, the flip-flop complements the current state. If M = 0, the next state of the flip-flop is equal to the value of N.
1. Derive the characteristic table for the flip-flop.
2. Show how a JK flip-flop can be converted to an MN flip-flop by adding gate(s) and inverter(s).
Hints and Answer List the steps necessary to read a word from memory in the 4 x 3 memory circuit shown in Figure 3.25.