1. BOOLEAN ALGEBRA Kamrul Ahsan Teacher of ICT @ITHS http://web.itu.edu.tr/~ahsan http://ahsan.bhaluka.net

2. Course Outline Boolean Algebra Relations Graphs Trees

3. Boolean Algebra Operation 1True 0False ∙ And +Or

4. Basic Law of Boolean Algebra 1 + 1 = 1, 1 + 0 = 1, 0 + 1 = 1, 0 + 0 = 0 1 ∙ 1 = 1, 1 ∙ 0 = 0, 0 ∙ 1 = 0, 0 ∙ 0 = 0

5. Example 1 F (x, y) = x ∙  y

6. Example 2 F (x, y) = xy +  z

7. Law of Boolean Algebra (1) x=x (1) Law of the double complement (2) Idempotent laws x + x=x x ∙ x =x

8. Law of Boolean Algebra (2) (3) Identity laws (4) Domination laws x + 1=1 x ∙ 0 =0 x + 0=x x ∙ 1 =x

9. Law of Boolean Algebra (3) (5) Commutative laws (6) Associative laws x + y=y + x x ∙ y = y ∙ x x + (y + z)= (x + y) + z x (yz) = (xy)z

10. Law of Boolean Algebra (4) (7) Distributive laws (8) De Morgan’s laws xy=  x +  y x + y =  x ∙  y x + (yz)= (x + y)(x + z) x (y + z) = xy + xz

11. Law of Boolean Algebra (5) (9) Absorption laws (10) Unit property x + xy= x x (x + y) = x x.  x= 0 (11) Zero property x +  x= 1

12. 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 F(x,y,z) = x  y zG(x,y,z) = x y  z +  x y  z

13. 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 =x  z + y  z =x 1  z + 1 y  z =x (y +  y)  z + (x +  x) y  z =xy  z + x  y  z + xy  z +  xy  z Distributive law Identity law Unit property Distributive law Idempotent law

14. Logic Gates AND gate Inverter OR gate x y x y x xy x + y xx

15. Combination of Gate (1) xy + xz x + xy x y xy xy + xz z xz x y x + xy xy x

16. Combination of Gate (2) x y x xx y xy  xy xy +  xy x y xx xy  xy xy +  xy

17. Example: combination of gate (x + y)  x  x (y +  z) (x + y + z)  x  y  z xy + xz + yz xy +  x  y xyz + x  y  z +  x y  z +  x  y z

18. Minimization of Circuits using laws (1) xyz + x  y z=(y +  y)(xz) =1 ∙ xz =xz x + x=(x + x) ∙ 1 =(x + x) ∙ (x +  x) =x + (x +  x) =x + 0 =x

19. 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 Unit property Distributive laws Identity laws Unit property