Logic Gates & Boolean Algebra

Chin-Sung Lin

Eleanor Roosevelt High School

Logic Gates & Boolean Algebra

• System Concept

• Systems & Subsystems

• Analog Systems vs. Digital Systems

• Combinational vs. Sequential Circuits

• Truth Tables & Basic Logic Gates

• Logic Simulation

• Digital Logic Circuits

• Equivalent Logic Circuits

• Digital Building Blocks

System Concept

What Are They In Common?

VS.

Computer Onion

System Concept

System(System Function)

Input Output

System Concept

System(System Function)

Input Output

Input

Output

System Function

System Concept

• A convenient way to view and understand both the nature and man-made worlds.

• Hide details and complexity from the viewers by encapsulating this detail information into a “system”.

• Only the inputs/outputs (I/O) of the system, and the function of the system (called system function) are important.

System(System Function)

Input Output

Systems & Subsystems

System(System Function)

Systems & Subsystems

• Systems can be further divided down to subsystems.

• Subsystems are connected together through inputs and outputs to form the larger system.

Input OutputSubsystem

Subsystem

Sub-system

Input

Input

Output

Subsystem

Subsystem

Sub-system

Input

Input

Output

Systems & Subsystems

• Subsystem themselves can also be further divided down to “Sub-subsystems”.

• This process can be continued until we reach the most basic elements of the digital logic world— basic logic gates.

• A system has a layered hierarchical structure like an onion.

Input OutputInput

Input

Output

Systems & Subsystems

Analog System vs. Digital Systems

Analog System vs. Digital Systems

• Two types of systems: Analog system and digital system.

AnalogSystem

Input Output

DigitalSystem

Input Output

Analog System vs. Digital Systems

• The values of input/output/internal signals of an analog system can vary over a continuous range of values.

AnalogSystem

Input Output

Analog System vs. Digital Systems

• The values of input/output/internal signals of a digital system can only be 1’s and 0’s.

DigitalSystem

Input Output0 10 01 0 101 1

Analog System vs. Digital Systems

• Lots of real-world signals are analog in nature.

• Analog-to-Digital (A/D) converter has been used to process (digitization/quantization) the incoming analog signals and change them to digital (binary) signals.

• The digital signals can now be processed by the digital system (e.g., microprocessor).

DigitalSystem

Digital I/P

0 10 01

A/DConverter

Analog I/P

Analog System vs. Digital Systems

• The output digital signals from the digital system are digital (binary) in nature.

• Digital-to-Analog (D/A) converter has been used to process the outgoing digital signals and change them to analog (continuous) signals.

• Most of the computers are the mix of analog and digital systems.

DigitalSystem

Digital I/P Digital O/P

0 10 01 0 101 1

D/AConverter

A/DConverter

Analog I/P Analog O/P

DigitalSystem

Digital I/P Digital O/P

0 10 01 0 101 1

D/AConverter

A/DConverter

Analog I/P Analog O/P

Analog System vs. Digital Systems

DigitalSystem

Digital I/P Digital O/P

0 10 01 0 101 1

D/AConverter

A/DConverter

Analog I/P Analog O/P

Digital Systems

• In this unit, we are going to focus ONLY on the digital system.

Combinational vs. Sequential Systems

Combinational vs. Sequential Systems

CombinationalSystemInput Output0 10 01 0 101 1

SequentialSystemInput Output0 10 01 0 101 1

• Two types of digital systems/circuits— combinational and sequential.

Combinational vs. Sequential Systems

• The outputs of a combinational digital system/circuits are solely decided by its inputs.

CombinationalSystemInput Output0 10 01 0 101 1

CombinationalSystemInput Output0 10 01 0 101 1

CombinationalSystemInput Output1 01 10 0 011 0

Combinational vs. Sequential Systems

• The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system.

SequentialSystemInput Output0 10 01 0 101 1

SequentialSystemInput Output0 10 01 1 010 0

SequentialSystemInput Output1 01 10 0 011 0

SequentialSystem

SequentialSystem

SequentialSystem

Combinational vs. Sequential Systems

• The outputs of a sequential digital system/circuits depend not only on its inputs, but also on the “current state” of the system.

• Sequential system/circuits have some sort of “memory” in it, so it can “memorize” the “current state” of the system and behave accordingly.

Input Output0 10 01 0 101 1

Input Output1 01 10 0 011 0

Input Output0 10 01 1 010 0

State: 101

State: 110

State: 001

Combinational vs. Sequential Systems

CombinationalSystemInput Output0 10 01 0 101 1

SequentialSystemInput Output0 10 01 0 101 1

SequentialSystemInput Output0 10 01 0 101 1

Combinational Systems

• In this unit, we are going to focus ONLY on the combinational digital system.

CombinationalSystemInput Output0 10 01 0 101 1

Truth Tables & Basic Logic Gates

Truth Tables

• A truth table shows how a logic circuit's output responds to various combinations of the inputs.

• A truth table describe the system function of a logic system.

• Use “1” and “0” to represent “T” and “F” respectively.

1 1

1

1

1

0 0 0

0

00

0

p

qpq

• Use a logic gate symbol to represent the function.

Truth Tables & Basic Logic Gates

X = AB

• There are eight basic logic gates. However, only AND, OR and NOT are the most fundamental ones.

Negations (NOT)

• The negation of a statement always has the opposite truth value of the original statement and is usually formed by adding the word not to the given statement.

• Statement Right angle is 90o

• Negation Right angle is not 90o

TRUE

FALSE

• Statement Triangle has 4 sides

• Negation Triangle does not have 4 sides

FALSE

TRUE

Truth Table – Negation (NOT)

• The relationship between a statement p and its negation ~p can be summarized in a truth table.

• A statement p and its negation ~p have opposite truth values.

p ~p

T F

F T

Conjunctions (AND)

• A compound statement formed by combining two simple statements using the word and.

• Statement: p, q • Conjunction: p and q • Symbols: p ^ q

Truth Table – Conjunctions (AND)

• A conjunction is true when both statements are true.• When one or both statements are false, the conjunction is false.

p q p ^ q

T T T

T F F

F T F

F F F

Disjunctions (OR)

• A compound statement formed by combining two simple statements using the word or.

• Statement: p, q • Disjunction p or q • Symbols: p V q

Truth Table – Disjunctions (OR)

• A disjunction is true when one or both statements are true.

• When both statements are false, the disjunction is false.

p q p V q

T T T

T F T

F T T

F F F

Truth Tables & Basic Logic Gates

• Use 1 and 0 to represent True and False respectively.

• The truth table summarizes all the possible values of input signals and their corresponding output signal values.

p ~p

T F

F T

X Y

1 0

0 1

Truth Tables & Basic Logic Gates

• The truth table is an effective way to describe the system function of a digital system.

Input X Output Y

X Y

1 0

0 1

0/1 1/0

System Function

Truth Tables & Basic Logic Gates

• A logic gate is a device performing a logical operation on one or more logical inputs, and produces a single logical output.

0/1 1/0

Logic Gate

X

Boolean Algebra & Boolean Functions

• Boolean algebra is a branch of algebra in which the values of the variables are the truth values true (1) and false (0).

• The main operations of Boolean algebra are the conjunction (and, ), the disjunction (or, +), and the negation (not, ).

0/1 1/0

Logic Gate

X

Boolean Algebra & Boolean Functions

• A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs.

Y = X Y = A + BY = A B

(NOT) (OR)(AND)

Truth Tables & Basic Logic Gates

XX Y

0 1

1 0

NOT(Inverter)

Boolean Function Truth Table Logic Gate

Y = X

Truth Tables & Basic Logic Gates

AND

Truth Table Logic Gate

A B Y

0 0 0

0 1 0

1 0 0

1 1 1

Y = A B

Boolean Function

Truth Tables & Basic Logic Gates

Truth Table Logic Gate

A B Y

0 0 0

0 1 1

1 0 1

1 1 1OR

Y = A + B

Boolean Function

Truth Tables & Basic Logic Gates

X Y

0 0

1 1

Buffer

Boolean Function Truth Table Logic Gate

Y = XX

Truth Tables & Basic Logic Gates

NAND

Truth Table Logic Gate

A B Y

0 0 1

0 1 1

1 0 1

1 1 0

Y = A B

Boolean Function

Truth Tables & Basic Logic Gates

Truth Table Logic Gate

A B Y

0 0 1

0 1 0

1 0 0

1 1 0NOR

Y = A + B

Boolean Function

Truth Tables & Basic Logic Gates

XOR

Truth Table Logic Gate

A B Y

0 0 0

0 1 1

1 0 1

1 1 0

Y = A ⊕ B

Boolean Function

Truth Tables & Basic Logic Gates

Truth Table Logic Gate

A B Y

0 0 1

0 1 0

1 0 0

1 1 1XNOR

Y = A B

Boolean Function

Y = A ⊕ B

Truth Tables & Basic Logic Gates

X = AB

Logic Simulation

Logic Simulation

• Logic simulation is the use of simulation software to predict the behavior of digital circuits.

Digital Logic Circuits

Digital Logic Circuits

• Logic gates can be connected and cascaded to form logic

circuits.

• Every logic circuits can be treated as a system, and can be

described by a system function— truth table.

• We can derive the truth table of the circuit by evaluating the

logic circuit stage by stage.

Y

Digital Logic Circuits

A B C D E F Y

0 0

0 1

1 0

1 1

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1

0 1 1

1 0 0

1 1 0

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1 1

0 1 1 0

1 0 0 1

1 1 0 0

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1 1 0

0 1 1 0 1

1 0 0 1 0

1 1 0 0 0

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1 1 0 0

0 1 1 0 1 0

1 0 0 1 0 1

1 1 0 0 0 0

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1 1 0 0 0

0 1 1 0 1 0 1

1 0 0 1 0 1 1

1 1 0 0 0 0 0

C

D

E

F

Y

Digital Logic Circuits

A B C D E F Y

0 0 1 1 0 0 0

0 1 1 0 1 0 1

1 0 0 1 0 1 1

1 1 0 0 0 0 0

C

D

E

F

Y = A B+

Equivalent Logic Circuits

Equivalent Logic Circuits

• Logic circuits of different gates and forms can have an identical truth table. These circuits are called equivalent logic circuits.

• This implies that a digital system with certain system function (truth table) can have many different implementations.

Laws of Boolean Algebra

One variableNOT: • x = xAND: • x · x = x• x · x = 0OR: • x + x = x• x + x = 1XOR: • x x = 0⊕• x x = 1⊕

Laws of Boolean Algebra

Commutativity• AND: x · y = y · x• OR: x + y = y + x• XOR: x y = y x⊕ ⊕

Associativity• AND: (x · y) · z = x · (y · z)• OR: (x + y) + z = x + (y + z)• XOR: (x y) z = x (y z)⊕ ⊕ ⊕ ⊕

Laws of Boolean Algebra

Distributivity• x · (y + z) = (x · y) + (x · z)• x + (y · z) = (x + y) · (x + z)• x · (y z) = (x · y) (x · z)⊕ ⊕

De Morgan’s laws• NAND: x · y = x + y• NOR: x + y = x · y

Logic Simplification

• We can apply the laws of Boolean algebra to reduce the expression to its simplest form (simplest defined as requiring the fewest gates to implement)

Equivalent Logic Circuits

Different implementations are chosen to meet different design considerations• Less number of gates (area)• less kind of gates (gate type), and • less stages of the circuits (speed).

Equivalent Logic Circuits

• For example, what are the following logic circuits?

Equivalent Logic Circuits

• For example, all of the following four logic circuits have the same truth table, and implement the XOR gate functionality.

Digital Building Blocks

Digital Building Blocks

• All these logic circuits can be encapsulated into a block (XOR gate) and treated as a system.

Digital Building Blocks

• This XOR gate can be further used as a building block to build larger and more complicated logic circuit such as a “full adder”.

Digital Building Blocks

• A full adder can again be treated as a building block to build a larger logic circuit called Arithmetic-Logic Unit (ALU).

Digital Building Blocks

• ALU can again be treated as a building block to build a Central Processing Unit (CPU).

Digital Building Blocks

• This process can go on and on to build an Intel® Intel® Core™ i7 Processor with these digital logic building blocks.

Intel Core i7 Processor

Digital Building Blocks

• The processor is so complicated at this level that it can contain hundreds of millions of basic logic gates.

Q & A