BOOLEAN ALGEBRA

Kamrul Ahsan

Teacher of ICT @ITHS

http://web.itu.edu.tr/~ahsan

http://ahsan.bhaluka.net

Course Outline

Boolean Algebra

Relations

Graphs

Trees

Boolean Algebra Operation

1 True

0 False

∙ And

+ Or

Basic Law of Boolean Algebra

x y x ∙ y x + y1 1 1 11 0 0 10 1 0 10 0 0 0

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

1 ∙ 1 = 1 , 1 ∙ 0 = 0 , 0 ∙ 1 = 0 , 0 ∙ 0 = 0

Example 1

F (x, y) = x ∙ y

x y `y x ∙ `y1 1 0 01 0 1 10 1 0 00 0 1 0

Example 2

F (x, y) = xy + z

x y z xy `z xy + `z

1 1 1 1 0 11 1 0 1 1 11 0 1 0 0 01 0 0 0 1 10 1 1 0 0 00 1 0 0 1 10 0 1 0 0 00 0 0 0 1 1

Law of Boolean Algebra (1)

x = x

(1) Law of the double complement

(2) Idempotent laws

x + x = xx ∙ x = x

Law of Boolean Algebra (2)

(3) Identity laws

(4) Domination laws

x + 1 = 1x ∙ 0 = 0

x + 0 = xx ∙ 1 = x

Law of Boolean Algebra (3)

(5) Commutative laws

(6) Associative laws

x + y = y + xx ∙ y = y ∙ x

x + (y + z) = (x + y) + z x (yz) = (xy)z

Law of Boolean Algebra (4)

(7) Distributive laws

(8) De Morgan’s laws

xy = x +yx + y = x ∙y

x + (yz) = (x + y)(x + z)

x (y + z) = xy + xz

Law of Boolean Algebra (5)

(9) Absorption laws

(10) Unit property

x + xy = xx (x + y) = x

x .x = 0(11) Zero property

x +x = 1

Example: find Boolean expression

Find Boolean expression that represent the functions F(x,y,z) and G(x,y,z) which are given in table

x y z F G1 1 1 0 01 1 0 0 11 0 1 1 01 0 0 0 00 1 1 0 00 1 0 0 10 0 1 0 00 0 0 0 0

F(x,y,z) = xy z G(x,y,z) = x yz +x yz

Example: find function expansion

Find function expansion for the function F(x,y,z) = (x + y)z and determine the function

F(x,y,z) = (x + y)z= xz + yz= x 1z + 1 yz= x (y +y)z + (x +x) yz= xyz + xyz + xyz + xyz

Distributive law

Identity law

Unit property

Distributive law

Idempotent law

Logic Gates

AND gate

Inverter

OR gate

x

y

x

y

x

xy

x + y

x

Combination of Gate (1)

xy + xz

x + xy

xy

xy

xy + xz

zxz

x

y

x + xy

xy

x

Combination of Gate (2)

xy

xx

y

xy

xy

xy + xy

xy

x

xy

xy

xy + xy

Example: combination of gate

(x + y)x

x (y +z)

(x + y + z)xyz

xy + xz + yz

xy + xy

xyz + xyz + x yz + xy z

Minimization of Circuits using laws (1)

xyz + xy z = (y +y)(xz)= 1 ∙ xz= xz

x + x = (x + x) ∙ 1= (x + x) ∙ (x +x)= x + (x +x)= x + 0= x

Minimization of Circuits using laws (2)

x + xy = x ∙ 1 + xy

= x (1 + y)

= x (y + 1)

= x ∙ 1

= x x + 1 = (x + 1) ∙ 1

= (x + 1) ∙ (x +x)

= x + 1 ∙x= x +x= 1

Identity laws

Distributive laws

Commutative laws

Domination laws

Identity laws

Identity laws

Unit property

Distributive laws

Identity laws

Unit property