Boolean Algebra
Discussion D6.1
Appendix G
Boolean Algebra andLogic Equations
• George Boole - 1854
• Boolean 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
Venn Diagrams and Minterms
x y
z
x
y zx
y y
x
x
z
x
y
z y
zx y
z
x yz
z
Venn Diagrams and Minterms
xyz + xyz + xyz = xz + xy
x y
z
x
y zx
y z y
x
x
z
x
y
z y
zx y
z
x yz