Boolean Algebra

Boolean Algebra, Logic

Gates

Jaimon Jacob,

Asst. Professor in CS

Department of Computer

Science

2x

4–2

Boolean Algebra

•Boolean algebra provides the operations and the rules for

working with the set {0, 1}.

•These are the rules that underlie electronic circuits, and the

methods we will discuss are fundamental to VLSI design.

•We are going to focus on three operations:

• Boolean complementation,

• Boolean sum, and

• Boolean product

4–3

Boolean Operations

•The complement is denoted by a bar (on the slides, we will use a minus sign). It is defined by

•-0 = 1 and -1 = 0.

•The Boolean sum, denoted by + or by OR, has the following values:

•1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0

•The Boolean product, denoted by or by AND, has the following values:

•1 1 = 1, 1 0 = 0, 0 1 = 0, 0 0 = 0

4–4

Boolean Functions and Expressions

•Definition: Let B = {0, 1}. The variable x is called a Boolean

variable if it assumes values only from B.

•A function from Bn, the set {(x1, x2, …, xn) |xiB,

1 i n}, to B is called a Boolean function of degree n.

•Boolean functions can be represented using expressions

made up from the variables and Boolean operations.

4–5


•The Boolean expressions in the variables x1, x2, …, xn are

defined recursively as follows:

• 0, 1, x1, x2, …, xn are Boolean expressions.

• If E1 and E2 are Boolean expressions, then (-E1),

(E1E2), and (E1 + E2) are Boolean expressions.

•Each Boolean expression represents a Boolean function. The

values of this function are obtained by substituting 0 and 1 for

the variables in the expression.

4–6


•For example, we can create Boolean expression in the

variables x, y, and z using the “building blocks”

0, 1, x, y, and z, and the construction rules:

•Since x and y are Boolean expressions, so is xy.

•Since z is a Boolean expression, so is (-z).

•Since xy and (-z) are expressions, so is xy + (-z).

•… and so on…

4–7


•Example: Give a Boolean expression for the Boolean function

F(x, y) as defined by the following table:

x y F(x, y)

0 0 0

0 1 1

1 0 0

1 1 0

•Possible solution: F(x, y) = (-x)y

4–8


•Another Example: •Possible solution I:

•F(x, y, z) = -(xz + y)

•0

•0

•1

•1

•F(x, y, z)

•1

•0

•1

•0

•z

•0•0

•1•0

•1•0

•0•0

•y•x

•0

•0

•0

•1

•1

•0

•1

•0

•1•1

•1•1

•0•1

•0•1

•Possible solution II:

•F(x, y, z) = (-(xz))(-y)

4–9


•There is a simple method for deriving a Boolean expression

for a function that is defined by a table. This method is based

on minterms.

•Definition: A literal is a Boolean variable or its complement. A

minterm of the Boolean variables x1, x2, …, xn is a Boolean

product y1y2…yn, where yi = xi or yi = -xi.

•Hence, a minterm is a product of n literals, with one literal for

each variable.

4–10


•Consider F(x,y,z) again: •F(x, y, z) = 1 if and only if:

•x = y = z = 0 or

•x = y = 0, z = 1 or

•x = 1, y = z = 0

•Therefore,

•F(x, y, z) =

(-x)(-y)(-z) +

(-x)(-y)z +

x(-y)(-z)

•0

•0

•1

•1

•F(x, y, z)

•1

•0

•1

•0

•z

•0•0

•1•0

•1•0

•0•0

•y•x

•0

•0

•0

•1

•1

•0

•1

•0

•1•1

•1•1

•0•1

•0•1

4–11

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4–13

Computers

• There are three different, but equally

powerful, notational methods

for describing the behavior of gates

and circuits

– Boolean expressions

– logic diagrams

– truth tables

4–14

Boolean algebra

• Boolean algebra: expressions in this

algebraic notation are an elegant and

powerful way to demonstrate the activity of

electrical circuits

4–15

4–16

• Logic diagram: a graphical

representation of a circuit

– Each type of gate is represented by a specific

graphical symbol

• Truth table: defines the function of a gate

by listing all possible input combinations

that the gate could encounter, and the

corresponding output

Truth Table

4–17

Gates

• Let’s examine the processing of the following six types of gates

– NOT

– AND

– OR

– XOR

– NAND

– NOR

4–18

4–19

4–20

NOT Gate

• A NOT gate accepts one input value

and produces one output value

Figure 4.1 Various representations of a NOT gate

4–21

NOT Gate

• By definition, if the input value for a NOT

gate is 0, the output value is 1, and if the

input value is 1, the output is 0

• A NOT gate is sometimes referred to as

an inverter because it inverts the input

value

4–22

AND Gate

• An AND gate accepts two input signals

• If the two input values for an AND gate are

both 1, the output is 1; otherwise, the

output is 0

Figure 4.2 Various representations of an AND gate

4–23


•Question: How many different Boolean functions of degree 1

are there?

•Solution: There are four of them, F1, F2, F3, and F4:

x F1 F2 F3 F4

0 0 0 1 1

1 0 1 0 1

4–24


•Question: How many different Boolean functions of degree 2

are there?

•Solution: There are 16 of them, F1, F2, …, F16:

•1

•0

•0

•0

•F2

•0

•0

•0

•0

•F1

•0•1•0

•1•0•1

•0•1•1

•0•0•0

•F3•y•x

•1

•1

•1

•0

•F8

•0

•1

•1

•0

•F7

•0

•0

•0

•1

•F9

•0

•0

•1

•0

•F5

•1

•1

•0

•0

•F4

•1

•0

•1

•0

•F6

•0

•1

•0

•1

•F1

1

•1

•0

•0

•1

•F1

0

•0

•1

•1

•1

•F1

2

•1

•0

•1

•1

•F1

4

•0

•0

•1

•1

•F1

3

•1

•1

•0

•1

•F1

5

•1

•1

•1

•1

•F1

6

4–25


•Question: How many different Boolean functions of degree n

are there?

•Solution:

•There are 2n different n-tuples of 0s and 1s.

•A Boolean function is an assignment of 0 or 1 to each of these

2n different n-tuples.

•Therefore, there are 22ndifferent Boolean functions.

4–26

Duality

•There are useful identities of Boolean expressions that can

help us to transform an expression A into an equivalent

expression B

•We can derive additional identities with the help of the dual of

a Boolean expression.

•The dual of a Boolean expression is obtained by

interchanging Boolean sums and Boolean products and

interchanging 0s and 1s.

4–27

Duality

•Examples:

•The dual of x(y + z) is •x + yz.

•The dual of -x1 + (-y + z) is •(-x + 0)((-y)z).

•The dual of a Boolean function F represented by a Boolean

expression is the function represented by the dual of this

expression.

•This dual function, denoted by Fd, does not depend on the

particular Boolean expression used to represent F.

4–28

Duality

•Therefore, an identity between functions represented by Boolean expressions remains valid when the duals of both sides of the identity are taken.

•We can use this fact, called the duality principle, to derive new identities.

•For example, consider the absorption law x(x + y) = x.

•By taking the duals of both sides of this identity, we obtain the equation x + xy = x, which is also an identity (and also called an absorption law).

4–29

4–30

4–31

OR Gate

• If the two input values are both 0, the

output value is 0; otherwise, the output is 1

Figure 4.3 Various representations of a OR gate

4–32

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Logic Gates

•Example: How can we build a circuit that computes the

function xy + (-x)y ?

xy + (-x)y

x

y

xy

x -x

y

(-x)y

4–34

4–35

4–36

XOR Gate

• XOR, or exclusive OR, gate

– An XOR gate produces 0 if its two inputs are

the same, and a 1 otherwise

– Note the difference between the XOR gate

and the OR gate; they differ only in one

input situation

– When both input signals are 1, the OR gate

produces a 1 and the XOR produces a 0

4–37

XOR Gate

Figure 4.4 Various representations of an XOR gate

NAND and NOR Gates

• The NAND and NOR gates are essentially the

opposite of the AND and OR gates, respectively

Figure 4.5 Various representations

of a NAND gate

Figure 4.6 Various representations

of a NOR gate

4–39

4–40

4–41

4–42

Gates with More Inputs

• Gates can be designed to accept three or more

input values

• A three-input AND gate, for example, produces

an output of 1 only if all input values are 1

Figure 4.7 Various representations of a three-input AND gate

4–43

3-Input And gate

A B C Y

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1

Y = A . B . C

4–44

Constructing Gates

• A transistor is a device that acts, depending on

the voltage level of an input signal, either as a

wire that conducts electricity or as a resistor that

blocks the flow of electricity

– A transistor has no moving parts, yet acts like

a switch

– It is made of a semiconductor material, which is

neither a particularly good conductor of electricity,

such as copper, nor a particularly good insulator,

such as rubber

4–45

Circuits

• Two general categories

– In a combinational circuit, the input values explicitly

determine the output

– In a sequential circuit, the output is a function of the

input values as well as the existing state of the circuit

• As with gates, we can describe the operations

of entire circuits using three notations

– Boolean expressions

– logic diagrams

– truth tables

4–46

Combinational Circuits

• Gates are combined into circuits by using the

output of one gate as the input for another

Page 99

AND

AND

OR

4–47

Combinational Circuits

• Because there are three inputs to this circuit, eight rows

are required to describe all possible input combinations

• This same circuit using Boolean algebra:

(AB + AC)

Page 100

4–48

Now let’s go the other way; let’s take a

Boolean expression and draw

• Consider the following Boolean expression: A(B + C)

Page 100

Page 101

• Now compare the final result column in this truth table to

the truth table for the previous example

• They are identical

4–49

Simple design problem

• A calculation has been done and its results

• are stored in a 3-bit number

• Check that the result is negative by anding

• the result with the binary mask 100

• Hint: a “mask” is a value that is anded with

• a value and leaves only the important bit

4–50

Now let’s go the other way; let’s take a

Boolean expression and draw

• We have therefore just demonstrated circuit

equivalence

– That is, both circuits produce the exact same output

for each input value combination

• Boolean algebra allows us to apply provable

mathematical principles to help us design

logical circuits

4–51

Adders

• At the digital logic level, addition is

performed in binary

• Addition operations are carried out

by special circuits called, appropriately,

adders

4–52

Adders

• The result of adding two binary digits could produce a carry value

• Recall that 1 + 1 = 10 in base two

• A circuit that computes the sum of two bits and produces the correct carry bit is called a half adder

• Notice the Sum & Carry are NEVER both 1.

(XOR) (AND)

4–53

Adders

• Circuit diagram

representing

a half adder

• Two Boolean

expressions:

sum = A B

carry = ABPage 103

4–54

Adders

• A circuit called a full adder takes the

carry-in value into account

Figure 4.10 A full adder

4–55

Adding Many Bits

• To add 2 8-bit values, we can duplicate a

full-adder circuit 8 times. The carry-out

from one place value is used as the carry

in for the next place value. The value of

the carry-in for the rightmost position is

assumed to be zero, and the carry-out of

the leftmost bit position is discarded

(potentially creating an overflow error).

4–56

4–57

as Universal Logic Gates

• Any logic circuit can be built using only NAND gates, or only NOR gates. They are the only logic gate needed.

• Here are the NAND equivalents:

NAND and NOR

4–58

NAND and NOR as Universal Logic Gates

(cont)

• Here are the NOR equivalents:

• NAND and NOR can be used to reduce the number of required gates in a circuit.

4–59

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4–61

Practice Assignment (check

the result)

4–62


•Definition: The Boolean functions F and G of n variables are

equal if and only if F(b1, b2, …, bn) = G(b1, b2, …, bn) whenever

b1, b2, …, bn belong to B.

•Two different Boolean expressions that represent the same

function are called equivalent.

•For example, the Boolean expressions xy, xy + 0, and xy1 are

equivalent.

4–63


•The complement of the Boolean function F is the function –F,

where –F(b1, b2, …, bn) =

-(F(b1, b2, …, bn)).

•Let F and G be Boolean functions of degree n. The Boolean

sum F+G and Boolean product FG are then defined by

•(F + G)(b1, b2, …, bn) = F(b1, b2, …, bn) + G(b1, b2, …, bn)

•(FG)(b1, b2, …, bn) = F(b1, b2, …, bn) G(b1, b2, …, bn)

Boolean Algebra

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