Boolean Algebra
Module M4.1
Section 5.1
Boolean Algebra andLogic Equations
• Switching Algebra Theorems
• Venn Diagrams
One-variable Theorems
OR Version AND Version
X # 0 = X
X # 1 = 1
X & 1 = X
X & 0 = 0
Note: Principle of Duality You can change # to & and 0 to 1 and vice versa
One-variable Theorems
OR Version AND Version
X # !X = 1
X # X = X
X & !X = 0
X & X = X
Note: Principle of Duality You can change # to & and 0 to 1 and vice versa
Two-variable Theorems
• Commutative Laws
• Unity
• Absorption-1
• Absorption-2
Commutative Laws
X # Y = Y # X
X & Y = Y & X
Venn Diagrams
X
!X
Venn Diagrams
X Y
X & Y
Venn Diagrams
X # Y
X Y
Venn Diagrams
!X & Y
X Y
Unity!X & Y
X Y
X & Y
(X & Y) # (!X & Y) = Y
Dual: (X # Y) & (!X # Y) = Y
Absorption-1
X Y
X & Y
Y # (X & Y) = Y
Dual: Y & (X # Y) = Y
Absorption-2!X & Y
X Y
X # (!X & Y) = X # Y
Dual: X & (!X # Y) = X & Y
Three-variable Theorems
• Associative Laws
• Distributive Laws
Associative Laws
X # (Y # Z) = (X # Y) # Z
Dual:
X & (Y & Z) = (X & Y) & Z
Associative Law
0 0 0 0 0 0 00 0 1 1 1 0 10 1 0 1 1 1 10 1 1 1 1 1 11 0 0 0 1 1 11 0 1 1 1 1 11 1 0 1 1 1 11 1 1 1 1 1 1
X Y Z Y # Z X # (Y # Z) X # Y (X # Y) # Z
X # (Y # Z) = (X # Y) # Z
Distributive Laws
X & (Y # Z) = (X & Y) # (X & Z)
Dual:
X # (Y & Z) = (X # Y) & (X # Z)
X Y
Z
X # (Y & Z) = (X # Y) & (X # Z)
Distributive Law - a
Distributive Law - b
X & (Y # Z) = (X & Y) # (X & Z)
X Y
Z
Generalized De Morgan’s Theorem
• NOT all variables
• Change & to # and # to &
• NOT the result
• --------------------------------------------
• F = X & Y # X & Z # Y & Z
• F = !((!X # !Y) & (!X # !Z) & (!Y # !Z))
• F = !(!(X & Y) & !(X & Z) & !(Y & Z))
X
Y
X
Z
Y
Z
F
F = !(!(X & Y) & !(X & Z) & !(Y & Z))
X
Y
X
Z
Y
Z
F
F = !(!(X & Y) & !(X & Z) & !(Y & Z))
NAND Gate
X Y
X Z
Y Z
F
F = X & Y # X & Z # Y & Z
Question
The following is a Boolean identity: (true or false) Y # (X & !Y) = X # Y