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Boolean Algebra
Discussion D6.1
Sections 13-3 – 13-6
Boolean Algebra andLogic Equations
• George Boole - 1854
• Switching Algebra Theorems
• Venn Diagrams
George BooleEnglish logician and mathematician
Publishes Investigation of theLaws of Thought in 1854
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
Question
The following is a Boolean identity: (true or false) Y | (X & ~Y) = X | Y
Absorption-2X & ~Y
Y X
Y | (X & ~Y) = X | Y