(3)Simplifying Logic Systems Using Boolean Algebra

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NAME:......

DATE ISSUED:......

COMPLETION DATE:......

SIMPLIFYING

DIGITAL

PROCESSOR

SUBSYSTEMS

PACKAGE

(1).Boolean Algebra.

(2).Combinational Logic.

(3)Simplifying Logic Systems using Boolean Algebra.

(4)Karnaugh Maps.

(5)Conversion to NAND and NOR gates.

(6)Describing the operation of combinations of logic gates.

(7)Additional Practical Work.

Boolean Algebra

The aim of this assignment is to introduce the concepts of Boolean Algebra.

You will need to use the Support Booklet to complete this assignment.

By the end of this task you will be able to:-

Recognise Boolean Algebra expressions,

Explain the meaning of a Boolean Algebra expression,

Recognise Boolean Identities.

1).Briefly explain what is meant by Boolean Algebra.

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2).If A and B are two logic inputs and Q is the logic output, state the Boolean expressions for:

(a)A AND B......

(b)A OR B......

(c)NOT A......

3).Explain the following Boolean expressions.

(a)AB......

(b)A + B......

(c)......

(d)......

(e)......

(f)......

(g)......

(h)......

Boolean Identities.

Boolean identities are basic Boolean expressions that can be directly simplified.

1).(a)Complete the list below. Those that are not obvious or that you cannot remember you should work out by drawing a truth table in the space provided.

(b)Prove, using truth tables, that:-

(i)

(ii)

(c)What is the name for these two identities?

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Combinational Logic.

The aim of this assignment is to refresh the concepts needed to produce Truth Tables for combinations of logic gates.

Use the Support Booklet to help you complete the following tasks.

By the end of this task you will be able to:-

Draw truth tables for basic combinations of logic gates,

Construct basic combinations of logic gates,

Work out Boolean expressions for these combinations.

1).The truth table for the circuit below is required. This can be determined in two completely different ways, the first being to actually construct the circuit and then determine the truth table experimentally.

(a)What type of logic gates are used in this circuit?......

(b)The 4011 or 4093 IC contains four of these logic gates. Complete the pin-out diagram below for these ICs.

(c)Mark onto the circuit diagram the pin numbers of the gates that you are going to use.

(d)Explain what should you do with the inputs of the logic gate that you are not using and why.

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(e)Draw onto the protoboard layout diagram below the IC and where you will connect wires to construct the circuit.

(f)A 1k resistor and a LED can be used to monitor each logic gate output.

Draw below how you would connect the resistor and LED so that the LED lit when the output is logic 1.

(g)Draw three such resistor / LED pairs on the circuit board diagram above to monitor the outputs of the three logic gates in the circuit, C, D, and E.

2).(a)Construct the circuit on your protoboard following the circuit board diagram that you have drawn above.

(b)Use your circuit to complete the truth table below for logic gate 1.

(c)What is the function of logic gate 1?

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(d)Use your circuit to complete the truth table below for logic gate 2.

(e)What is the function of logic gate 2 and state the Boolean expression for this gate?

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(f)Now use your circuit to complete the truth table below for the whole circuit.

(g)What single logic gate can be used to replace the three logic gates in the circuit and produce the same truth table?

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(h)State the Boolean expression for the whole circuit and show what it simplifies to.

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(i)Explain how you would derive the truth table for a logic gate circuit without constructing it first.

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3).The circuit diagram below is quite common. Determine the truth table for the circuit by following the same process as for the previous circuit.

(a)Sketch the pin out diagram for the 4011 / 4093 ICs

(b)Mark onto the circuit diagram the pin connections for the arrangement of logic gates that you are going to use.

(c)Draw onto the protoboard layout diagram below the IC and where you will connect wires to construct the circuit.

(d)Add to your layout diagram four resistors and LEDs to monitor the output of the logic gates as you did with the previous example.

(e)Use the circuit that you have constructed to complete the truth table below.

(f)What single logic gate can be used to replace the four logic gates in the circuit and produce the same truth table?

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(g)Draw the circuit symbol for this logic gate in the space below.

(h)State the Boolean expression for this logic function.

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Combinational Logic Examples.

The aim of this assignment is to provide further opportunities to practice producing Truth Tables for combinations of Logic gates.

Use the Foundation Support Booklet to help you complete the following tasks.

4).(a)Draw the Truth Table for the logic circuit in the space below.

Remember that with three inputs there will be 23 = 8 possible input combinations.

(b)State the Boolean expression for the circuit above.

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5).(a).Identify the types of logic gates marked 1 and 2 in the diagram opposite.

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(b).Draw below a truth table to describe the operation of the logic circuit.

(c)State the Boolean expression for the logic circuit.

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(d)In the logic circuit, A and B are connected to two sensors and Q is connected to a buzzer.

A is connected to a light sensor which gives logic 1 in the light and a logic 0 in the dark.

B is connected to a temperature sensor which gives a logic 1 when cold and a logic 0 when hot.

Q is connected to a buzzer which sounds when driven by a logic 1.

What are the input conditions of the light and temperature sensors that will cause a buzzer to sound?

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3).The law states that drivers and passengers in a car must wear seat belts when the car is moving. A system is to be designed to make a buzzer sound if the seat belt has not been fastened but the seat is sat on. Each seat has in it a pressure pad which gives a logic 1 only when sat on.

Each belt has a switch in it that gives a logic 1 only when fastened.

(a)Draw a Truth table to describe when the when the buzzer should sound.

Use PRESSURE PAD, BELT SWITCH, and BUZZER as the truth table headings.

Write 1 when the buzzer should sound and 0 when it should be silent.

(b)(i)Draw a circuit diagram below, using logic gates, to show how the system could be made.

(ii)Explain how your circuit operates.

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4).a).Complete the truth table below:

INPUT / OUTPUT
A / B / AND gate / NAND gate
0 / 0 / 0 / 1
0 / 1
1 / 0
1 / 1

(3)

(b)"Logic Gates are digital devices".

What is meant by "digital" as used in the above sentence?

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(2)

c).An OR gate is connected as shown opposite. At the start the voltage signal levels at A and B are low (0)

The signal level at A is changed as shown in the table below.

Complete this Truth Table.

INPUT / OUTPUT
A / B / C
start / 0 / 0 / 0
1st change / 1
2nd change / 0
3rd change / 1

(3)

(d)Input A now returns to low (0) and switch S is closed.

What will happen to the output C?

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(2)

(e)State a use for this circuit.

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(1)

5).(a)Draw the symbol for an inverter.

(1)

(b)Below are the truth tables for two logic gates, C and D.

INPUT / OUTPUT
A / B / Logic gate C / Logic gate D
0 / 0 / 0 / 1
0 / 1 / 1 / 1
1 / 0 / 1 / 1
1 / 1 / 1 / 0

(i)What is the correct name for gate C?......

(ii)What is the correct name for gate D?......

(2)

(c)When it rains, car drivers are supposed to put their headlights on.

A light sensor is placed in front of one headlamp.

A rain sensor is placed on the bonnet.

The system below must make a buzzer in the car sound when it rains and the headlamps are off.

(i)Use the above information to complete the truth table for the logic gate E needed in this system.

INPUTS TO E / OUTPUT FROM
Light sensor / Rain sensor / E TO BUZZER
0 (headlamp off) / 0 (wet)
0 (headlamp off) / 1 (dry)
1 (headlamp on) / 0 (wet)
1 (headlamp on) / 1 (dry)

(4)

(ii)What is the correct name for gate E?......

(1)

Simplifying Logic Systems using Boolean Algebra.

The aim of this assignment is to provide practice at applying Boolean identities to logic systems in order to simplify them. You will need to use the Support Booklet.

1).Simplify the following Boolean expressions. Use the space below each expression for deriving your answer.

(a) B

(b) B bar

(d) C bar

(e) A+B

(f) A+B

(g) A+C

(h) B

Karnaugh Maps.

The aim of this assignment is to understand Karnaugh maps and how they can be constructed from truth tables and logic expressions. You will need to use the Support Booklet.

1).(a)Draw the truth table for a two input AND gate and the Karnaugh map by the side.

Truth tableKarnaugh map

(b)Draw the truth table for a three input OR gate and the Karnaugh map by the side.

Truth tableKarnaugh map

(c)Draw the truth table for a four input NAND gate and the Karnaugh map by the side.

Truth tableKarnaugh map

Simplifying logic systems using Karnaugh maps.

2).(a)Draw a Karnaugh map to represent the Boolean expression

(b)Explain how a Karnaugh map can be used to simplify a Boolean expression.

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(c)Use the Karnaugh map to simplify the expression in part (a) and show that your answer agrees with the simplification from using Boolean algebra.

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(d)Draw a Karnaugh map to represent the Boolean expression

(e)Use the Karnaugh map to simplify this expression in and show that your answer agrees with the simplification from using Boolean algebra.

(f)Draw a Karnaugh map for the logic expression

(g)Simplify the expression in part (f) using the Karnaugh map.

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(h)Draw a Karnaugh map for the logic expression

(i)Simplify the expression in part (g) using the Karnaugh map.

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Converting logic systems to NOR and NAND gates only

The aim of this assignment is to gain practice at converting logic systems to NOR and NAND gates only. You will need to use the Support Booklet.

1).(a)Using DeMorgan's theorem, show that an AND gate can be constructed from only NOR gates.

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(b)Draw the circuit diagram of the arrangement.

(c)Using DeMorgan's theorem, show that an OR gate can be constructed from only NAND gates.

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(b)Draw the circuit diagram of the arrangement.

2).(a)Use DeMorgan's theorem to convert this expression into one having only NAND and NOR gates.

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(b)Draw the circuit diagram for this NAND and NOR solution.

(c)Draw the circuit diagram for the Boolean expression below:

(d)Use DeMorgan's theorem to change this into a circuit using only NAND gates.

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(e)Draw the circuit diagram of the new arrangement.

(f)State the name of the logic gate with this function.

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3).(a)Draw the circuit diagram for the Boolean expression:

(b)Use DeMorgan's theorem to convert it into a circuit using only NAND gates.

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(c)Draw the circuit diagram using only NAND gates.

Describing the operation of combinational logic systems.

The aim of this assignment is to gain practice describing the operation and function of a combinational logic system from the circuit diagram. You will need to use the Support Booklet.

1).(a)Derive the Boolean expression for the circuit below.

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(b)Derive the Boolean expression for each of the outputs for the circuit below.

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(c)Derive the Boolean expression for the circuit below:

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(d)Derive the Boolean expression for the circuit below:

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Additional Practical Work.

The aim of this assignment is to provide additional practical work in determining the operation and truth table for a logic gate circuit.

1).The following circuit is part of an electronic combination lock. If the three switches A, B and C are closed in an incorrect sequence an alarm will be activated (logic 1).

(a)The IC that you need is a 4001. Complete the pinout for the IC in the space below.

(b)Construct the circuit on protoboard and investigate the logic state of the output for all possible combinations of the inputs A, B and C. Summarise your results into the table below.

INPUT A / INPUT B / INPUT C / OUTPUT
0 / 0 / 0
0 / 0 / 1
0 / 1 / 0
0 / 1 / 1
1 / 0 / 0
1 / 0 / 1
1 / 1 / 0
1 / 1 / 1

(c)Complete the truth table below to check your experimental results.

INPUT A / INPUT B / INPUT C / OUTPUT
0 / 0 / 0
0 / 0 / 1
0 / 1 / 0
0 / 1 / 1
1 / 0 / 0
1 / 0 / 1
1 / 1 / 0
1 / 1 / 1

(d)What is the order of closure of the switches in order not to activate the alarm?

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