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