course name: discrete structures course code: am103 batch

Course Name: Discrete Structures

Course Code: AM103

Batch 2020

BE(CSE)

Department of Applied Sciences

9/16/2021 Department of Applied Sciences,HP 1

Boolean Algebra and its

application to switching

circuits


Definition

BOOLEAN ALGEBRA :

Definition : A Boolean Algebra is a distributive,

complemented lattice having at least two distinct elements

as zero element 0 and a one element 1.


Definition


Definition

Boolean Algebra

Let B be a set with two binary operations ˅ and ˄ or + and *

and two elements 0 and 1 and a unary operation ‘–’ such that

the following properties holds:

1. Closure Property i.e. a+b a.b a,b

2. Commutative Property i.e. a+b=b+a and a.b=b.a a,b

3. Identity Law i.e. a+0=a,a.1=a a

4. Distributive Law i.e. a+(b.c)=(a+b).(a+c) a,b,c

5. Complement Law i.e. a+a=1 and a.a =0 a

6. Associative Law i.e. a+(b+c)= (a+b)+c

and a.(b.c)=(a.b).c a,b,c


Dual of Boolean Expression

• The principle of duality is an important concept. This says that

if an expression is valid in Boolean algebra, the dual of that

expression is also valid.

• To form the dual of an expression, replace all (+) operators with

(·) operators, all (·) operators with (+) operators, all ones with

zeros, and all zeros with ones.

• Following the replacement rules…

a(b + c) = ab + ac

• Form the dual of the expression

a + (bc) = (a + b)(a + c)

• Take care not to alter the location of the parentheses if they are

present.

6

Examples


TRUTH TABLES

A truth table is a mathematical table used in logic—

specifically in connection with Boolean algebra, Boolean

functions, and propositional calculus—which sets out the

functional values of logical expressions on each of their

functional arguments, that is, for each combination of values

taken by their logical variables.


DeMorgan’s Theorem

Prove (x + y)’ = x’y’ and (xy)’ = x’ + y’

Using truth table

x y x’ y’ x+y (x+y)’ x’y’ xy x’+y' (xy)’

0 0 1 1 0 1 1 0 1 1

0 1 1 0 1 0 0 0 1 1

1 0 0 1 1 0 0 0 1 1

1 1 0 0 1 0 0 1 0 0

9

Example


Example

Show that xy + x’z + yz = xy + x’z using Boolean

identity laws

• Proof:

– xy + x’z + yz

• = xy + x’z + 1.yz

• = xy + x’z + (x+x’)yz

• = xy + x’z + xyz + x’yz

• = (xy + xyz) + (x’z + x’zy)

• = x(y + yz) + x’ (z + zy)

• = xy + x’z

11

Example

Simplify this expression using Boolean identities

x(x'+y)

solution

x(x'+y)

= xx' + xy

= 0+xy

= xy


Logic Gates

Digital logic gates are the building blocks from which all

digital electronic circuits and microprocessor based systems

are made


A

B

Z

𝐙 = 𝐀 + 𝐁

OR GATE

A

BZ

𝐙 = 𝐀●𝐁

AND GATE

A Z

𝐙 = ഥ𝐀

NOT GATE

A

B

Z

𝐙 = 𝐀 + 𝐁 = ഥ𝐀●ഥ𝐁

NOR GATE

A

B

Z

𝐙 = 𝐀●𝐁 = ഥ𝐀 + ഥ𝐁

NAND GATE

NAND and NOR Gates are known as Universal

gates because any digital circuit can be realized

completely by using either of these two gates and

also they can generate the three basic gates AND,

OR and NOT Gate

A universal gate provides flexibility and offers

enormous advantage to logic designers.

UNIVERSAL GATES

A

B

Z

XOR (EXCLUSIVE OR GATE)

AB

Z

XNOR GATE(EXCLUSIVE NOR GATE)

Draw the circuit diagram of the following

Boolean function

𝒇𝟏 𝒙, 𝒚, 𝒛 = 𝒙𝒚ത𝒛

x

y

z

Draw the circuit diagram of the following

Boolean function

𝒇𝟐 𝒙, 𝒚, 𝒛 = 𝒙 + ഥ𝒚𝒛

x

yz

Practice Questions


course name: discrete structures course code: am103 batch

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