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