ece –digital electronics

ECE – Digital Electronics

Basic Logic Operations,Boolean Expressions,Boolean Expressions,

andBoolean Algebra

Digital Electronics

Basic Logic Operations,Boolean Expressions,Boolean Expressions,

andBoolean Algebra

Basic Logic Operations

ECE Digital Electronics

Basic Logic Operations

ECE Digital Electronics 2

Basic Logic Operations

� AND

� OR

� NOT (Complement)

ECE - Digital Electronics

� Order of Precedence

1. NOT

2. AND

3. OR

− can be modified using parenthesis

Basic Logic Operations

Digital Electronics 3

can be modified using parenthesis

Basic Logic Operations

ECE - Digital Electronics

Basic Logic Operations

Digital Electronics 4

Basic Logic Operations

ECE - Digital Electronics

Basic Logic Operations

Digital Electronics 5

Additional Logic Operations

� NAND

− F = (A . B)'

� NOR

− F = (A + B)'

ECE - Digital Electronics

− F = (A + B)'

� XOR

− Output is 1 iff either input is 1, but not both.

� XNOR (aka. Equivalence)

− Output is 1 iff both inputs are 1 or both inputs are 0.

Additional Logic Operations

Digital Electronics 6

Output is 1 iff either input is 1, but not both.

XNOR (aka. Equivalence)

Output is 1 iff both inputs are 1 or both inputs

Additional Logic OperationsNAND

ECE - Digital Electronics

NOR denotes inversion

Additional Logic OperationsXOR

Digital Electronics 7

XNORdenotes inversion

Exercise:

Derive the Truth table for each of the

Additional Logic Operations

ECE - Digital Electronics

Derive the Truth table for each of the following logic operations:

1. 2-input NAND2. 2-input NOR

Exercise:

Derive the Truth table for each of the

Additional Logic Operations

Digital Electronics 8

Derive the Truth table for each of the following logic operations:

input NANDinput NOR

Exercise:

Derive the Truth table for each of the

Additional Logic Operations

ECE - Digital Electronics

Derive the Truth table for each of the following logic operations:

1. 2-input XOR2. 2-input XNOR

Exercise:

Derive the Truth table for each of the

Additional Logic Operations

Digital Electronics 9

Derive the Truth table for each of the following logic operations:

input XORinput XNOR

Truth Tables

ECE - Digital Electronics

Truth Tables

Digital Electronics 10

Truth Tables� Used to describe the functional behavior of a Boolean

expression and/or Logic circuit.

� Each row in the truth table represents a unique combination of the input variables.

− For n input variables, there are 2

ECE - Digital Electronics

− For n input variables, there are 2

� The output of the logic function is defined for each row.

� Each row is assigned a numerical value, with the rows listed in ascending order.

� The order of the input variables defined in the logic function is important.

Truth TablesUsed to describe the functional behavior of a Boolean expression and/or Logic circuit.

Each row in the truth table represents a unique combination of the input variables.

For n input variables, there are 2n rows.

Digital Electronics 11

For n input variables, there are 2 rows.

The output of the logic function is defined for each

Each row is assigned a numerical value, with the rows

The order of the input variables defined in the logic

3-input Truth Table

F(A,B,C) = Boolean expression

ECE - Digital Electronics

F(A,B,C) = Boolean expression

input Truth Table

F(A,B,C) = Boolean expression

Digital Electronics 12

F(A,B,C) = Boolean expression

4-input Truth Table

F(A,B,C,D) = Boolean expression

ECE - Digital Electronics

F(A,B,C,D) = Boolean expression

input Truth Table

F(A,B,C,D) = Boolean expression

Digital Electronics 13

F(A,B,C,D) = Boolean expression

Boolean Expressions

ECE - Digital Electronics

Boolean Expressions

Digital Electronics 14

Boolean Expressions

� Boolean expressions are composed of

� Literals – variables and their complements

� Logical operations

� Examples

ECE - Digital Electronics

� Examples

� F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C'

� F = (A+B+C').(A'+B'+C).(A+B+C)

� F = A.B'.C' + A.(B.C' + B'.C)

literals

Boolean Expressions

Boolean expressions are composed of

variables and their complements

Digital Electronics 15

F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C'

F = (A+B+C').(A'+B'+C).(A+B+C)

F = A.B'.C' + A.(B.C' + B'.C)

logic operations

Boolean Expressions

� Boolean expressions are realized using a network (or combination) of logic gates.

− Each logic gate implements one of the logic operations in the Boolean expression

− Each input to a logic gate represents one of

ECE - Digital Electronics

− Each input to a logic gate represents one of the literals in the Boolean expression

A

B

literals

Boolean Expressions

Boolean expressions are realized using a network (or combination) of logic gates.

Each logic gate implements one of the logic operations in the Boolean expression

Each input to a logic gate represents one of

Digital Electronics 16

Each input to a logic gate represents one of the literals in the Boolean expression

f

logic operations

Boolean Expressions

� Boolean expressions are evaluated by

� Substituting a 0 or 1 for each literal

� Calculating the logical value of the expression

A Truth Table specifies the value of the Boolean

ECE - Digital Electronics

� A Truth Table specifies the value of the Boolean expression for every combination of the variables in the Boolean expression.

� For an n-variable Boolean expression, the truth table has 2n rows (one for each combination).

Boolean Expressions

Boolean expressions are evaluated by

Substituting a 0 or 1 for each literal

Calculating the logical value of the expression

specifies the value of the Boolean

Digital Electronics 17

specifies the value of the Boolean expression for every combination of the variables in the Boolean expression.

variable Boolean expression, the truth rows (one for each combination).

Boolean Expressions

Example:

Evaluate the following Boolean expression, for all combination of inputs, using a Truth

ECE - Digital Electronics

for all combination of inputs, using a Truth table.

F(A,B,C) = A'.B'.C + A.B'.C' + A.C

Boolean Expressions

Example:

Evaluate the following Boolean expression, for all combination of inputs, using a Truth

Digital Electronics 18

for all combination of inputs, using a Truth table.

F(A,B,C) = A'.B'.C + A.B'.C' + A.C

Boolean Expressions

� Two Boolean expressions are equivalent if they have the same value for each combination of the variables in the Boolean expression.

− F1

= (A + B)'

ECE - Digital Electronics

1

− F2

= A'.B'

� How do you prove that two Boolean expressions are equivalent?

− Truth table

− Boolean Algebra

Boolean Expressions

Two Boolean expressions are equivalent if they have the same value for each combination of the variables in the Boolean expression.

Digital Electronics 19

How do you prove that two Boolean expressions are equivalent?

Boolean Expressions

Example:

Using a Truth table, prove that the following two Boolean expressions are equivalent.

ECE - Digital Electronics

two Boolean expressions are equivalent.

F1

= (A + B)'F

2= A'.B'

Boolean Expressions

Example:

Using a Truth table, prove that the following two Boolean expressions are equivalent.

Digital Electronics 20

two Boolean expressions are equivalent.

= (A + B)'= A'.B'

Boolean Algebra

ECE - Digital Electronics

Boolean Algebra

Digital Electronics 21

Boolean Algebra� George Boole developed an algebraic description for

processes involving logical thought and reasoning.

− Became known as Boolean Algebra

� Claude Shannon later demonstrated that Boolean Algebra could be used to describe switching circuits.

ECE - Digital Electronics

Algebra could be used to describe switching circuits.

− Switching circuits are circuits built from devices that switch between two states (e.g. 0 and 1).

− Switching Algebra is a special case of Boolean Algebra in which all variables take on just two distinct values

� Boolean Algebra is a powerful tool for analyzing and designing logic circuits.

Boolean AlgebraGeorge Boole developed an algebraic description for processes involving logical thought and reasoning.

Boolean Algebra

Claude Shannon later demonstrated that Boolean Algebra could be used to describe switching circuits.

Digital Electronics 22

Algebra could be used to describe switching circuits.

Switching circuits are circuits built from devices that switch between two states (e.g. 0 and 1).

is a special case of Boolean Algebra in which all variables take on just two distinct

Boolean Algebra is a powerful tool for analyzing and

Basic Laws and Theorems

Commutative Law A + B = B + A

Associative Law A + (B + C) = (A + B) + C

Distributive Law A.(B + C) = AB + AC

Null Elements A + 1 = 1

Identity A + 0 = A

A + A = AIdempotence

ECE - Digital Electronics

A + A = A

Complement A + A' = 1

Involution A'' = A

Absorption (Covering) A + AB = A

Simplification A + A'B = A + B

DeMorgan's Rule (A + B)' = A'.B'

Logic Adjacency (Combining) AB + AB' = A

Consensus AB + BC + A'C = AB + A'C

Idempotence

Basic Laws and Theorems

A.B = B.A

A + (B + C) = (A + B) + C A . (B . C) = (A . B) . C

A + (B . C) = (A + B) . (A + C)

A . 0 = 0

A . 1 = A

A . A = A

Digital Electronics 23

A . A = A

A . A' = 0

A . (A + B) = A

A . (A' + B) = A . B

(A . B)' = A' + B'

(A + B) . (A + B') = A

AB + BC + A'C = AB + A'C (A + B) . (B + C) . (A' + C) = (A + B) . (A' + C)

Idempotence

A + A = A

F = ABC + ABC' + ABC

F = ABC + ABC'

Note: terms can also be added using this theorem

ECE - Digital Electronics

A . A = A

G = (A' + B + C').(A + B' + C).(A + B' + C)

G = (A' + B + C') + (A + B' + C)

Note: terms can also be added using this theorem

Idempotence

A + A = A

F = ABC + ABC' + ABC

F = ABC + ABC'

Note: terms can also be added using this theorem

Digital Electronics 24

A . A = A

G = (A' + B + C').(A + B' + C).(A + B' + C)

G = (A' + B + C') + (A + B' + C)

Note: terms can also be added using this theorem

Complement

A + A' = 1

F = ABC'D + ABCD

F = ABD.(C' + C)

F = ABD

ECE - Digital Electronics

A . A' = 0

G = (A + B + C + D).(A + B' + C + D)

G = (A + C + D) + (B . B')

G = A + C + D

Complement

A + A' = 1

F = ABC'D + ABCD

F = ABD.(C' + C)

F = ABD

Digital Electronics 25

A . A' = 0

G = (A + B + C + D).(A + B' + C + D)

G = (A + C + D) + (B . B')

G = A + C + D

Distributive Law

A.(B + C) = AB + AC

F = WX.(Y + Z)

F = WXY + WXZ

ECE - Digital Electronics

G = B'.(AC + AD)

G = AB'C + AB'D

H = A.(W'X + WX' + YZ)

H = AW'X + AWX' + AYZ

Distributive Law

A + (B.C) = (A + B).(A + C)

F = WX + (Y.Z)

F = (WX + Y).(WX + Z)

Digital Electronics 26

G = B' + (A.C.D)

G = (B' + A).(B' + C).(B' + D)

H = A + ( (W'X).(WX') )

H = (A + W'X).(A + WX')

Absorption (Covering)

A + AB = A

F = A'BC + A'

F = A'

ECE - Digital Electronics

G = XYZ + XY'Z + X'Y'Z' + XZ

G = XYZ + XZ + X'Y'Z'

G = XZ + X'Y'Z'

H = D + DE + DEF

H = D

Absorption (Covering)

A.(A + B) = A

F = A'.(A' + BC)

F = A'

Digital Electronics 27

G = XZ.(XZ + Y + Y')

G = XZ.(XZ + Y)

G = XZ

H = D.(D + E + EF)

H = D

Simplification

A + A'B = A + B

F = (XY + Z).(Y'W + Z'V') +

F = Y'W + Z'V' +

ECE - Digital Electronics

A.(A' + B) = A . B

G = (X + Y).( (X + Y)'

G = (X + Y) .

Simplification

A + A'B = A + B

.(Y'W + Z'V') + (XY + Z)'

F = Y'W + Z'V' + (XY + Z)'

Digital Electronics 28

A.(A' + B) = A . B

(X + Y)' + (WZ) )

(X + Y) . WZ

Logic Adjacency (Combining)

A.B + A.B' = A

F = (X + Y).(W'X'Z)

F = (X + Y)

ECE - Digital Electronics

(A + B).(A + B') = A

G = (XY + X'Z').(XY +

G = XY

Logic Adjacency (Combining)

A.B + A.B' = A

(W'X'Z) + (X + Y).(W'X'Z)'

F = (X + Y)

Digital Electronics 29

(A + B).(A + B') = A

).(XY + (X'Z')' )

G = XY

Boolean Algebra

Example:

Using Boolean Algebra, simplify the following Boolean expression.

ECE - Digital Electronics

Boolean expression.

F(A,B,C) = A'.B.C + A.B'.C + A.B.C

Boolean Algebra

Example:

Using Boolean Algebra, simplify the following Boolean expression.

Digital Electronics 30

Boolean expression.

F(A,B,C) = A'.B.C + A.B'.C + A.B.C

Boolean Algebra

Example:

Using Boolean Algebra, simplify the following

ECE - Digital Electronics

Using Boolean Algebra, simplify the following Boolean expression.

F(A,B,C) = (A'+B'+C').(A'+B+C').(A+B'+C')

Boolean Algebra

Example:

Using Boolean Algebra, simplify the following

Digital Electronics 31

Using Boolean Algebra, simplify the following Boolean expression.

F(A,B,C) = (A'+B'+C').(A'+B+C').(A+B'+C')

DeMorgan's Laws

� Can be stated as follows:

− The complement of the product (AND) is the sum (OR) of the complements.

� (X.Y)' = X' + Y'

ECE - Digital Electronics

− The complement of the sum (OR) is the product (AND) of the complements.

� (X + Y)' = X' . Y'

� Easily generalized to n variables.

� Can be proven using a Truth table

DeMorgan's Laws

Can be stated as follows:

The complement of the product (AND) is the sum (OR) of the complements.

(X.Y)' = X' + Y'

Digital Electronics 32

The complement of the sum (OR) is the product (AND) of the complements.

(X + Y)' = X' . Y'

Easily generalized to n variables.

Can be proven using a Truth table

Proving DeMorgan's Law

(X . Y)' = X' + Y'

ECE - Digital Electronics

Proving DeMorgan's Law

(X . Y)' = X' + Y'

Digital Electronics 33

x 1

x 2

x 1

x 2

x 1 x 2 = (a)

DeMorgan's Theorems

ECE - Digital Electronics

x 1

x 2

x 1

x 2

1 2

x 1 x 2 + (b)

x 1

x 2

x 1 x 2 + =

DeMorgan's Theorems

Digital Electronics 34

x 1

x 2

1 2

2 x 1 x 2 =

Importance of Boolean Algebra

� Boolean Algebra is used to simplify Boolean expressions.

– Through application of the Laws and Theorems discussed

ECE - Digital Electronics

discussed

� Simpler expressions lead to simpler circuit realization, which, generally, reduces cost, area requirements, and power consumption.

� The objective of the digital circuit designer is to design and realize optimal digital circuits.

Importance of Boolean Algebra

Boolean Algebra is used to simplify Boolean

Through application of the Laws and Theorems

Digital Electronics 35

Simpler expressions lead to simpler circuit realization, which, generally, reduces cost, area requirements, and

The objective of the digital circuit designer is to design and realize optimal digital circuits.

Algebraic Simplification

� Justification for simplifying Boolean expressions:

– Reduces the cost associated with realizing the expression using logic gates.

– Reduces the area (i.e. silicon) required to fabricate the switching function.

ECE - Digital Electronics

switching function.

– Reduces the power consumption of the circuit.

� In general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or minimum number of literals.

– No unique solution

Algebraic Simplification

Justification for simplifying Boolean expressions:

Reduces the cost associated with realizing the expression using logic gates.

Reduces the area (i.e. silicon) required to fabricate the

Digital Electronics 36

Reduces the power consumption of the circuit.

In general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or minimum number of literals.

Algebraic Simplification

� Boolean (or Switching) expressions can be simplified using the following methods:

1. Multiplying out the expression

2. Factoring the expression

ECE - Digital Electronics

3. Combining terms of the expression

4. Eliminating terms in the expression

5. Eliminating literals in the expression

6. Adding redundant terms to the expression

As we shall see, there are other tools that can be used to simplify Boolean Expressions.Namely, Karnaugh Maps.

Algebraic Simplification

Boolean (or Switching) expressions can be simplified using the following methods:

Multiplying out the expression

Factoring the expression

Digital Electronics 37

Combining terms of the expression

Eliminating terms in the expression

Eliminating literals in the expression

Adding redundant terms to the expression

As we shall see, there are other tools that can be used to simplify Boolean Expressions.