Boolean algebra

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets. It is also a special case of De Morgan algebra and Kleene algebra.



Reference:
http://en.wikipedia.org/wiki/Boolean_algebra_%28structure%29
http://zekiakan.tripod.com/boole_cebiri.htm

Absorption

• Absorption:

A) x + xy = x
B) x (x+y) = x



AXIOMS AND BASIC THEOREMS
Absorption x+x·y =x   x · (x + y) = x



A Boolean function that is expressed in algebraic form can be simpli-
fied using the axioms and theorems of Boolean algebra

⊳ Example 1: OR form of the Absorption Theorem

x + x · y = x · 1 + x · y = x · (1 + y) = x · 1 = x

⊳ Example 2:

x+x·y =x+x·y+x·y
=x·x+x·y+x·x+x·y
= (x + x) · (x + y) = 1 · (x + y)
=x+y

⊳ Example 3:

x · (x + y) = x · x + x · y = x + x · y = x



https://docs.google.com/viewer?a=v&q=cache:vvf4G4lWnnAJ:www.utdallas.edu/~cantrell/ee2310/boole.pdf+&hl=en&pid=bl&srcid=ADGEESj4MkhQetdjm9j72t0ctUL72kuNtjfsvEj1V1PBxgT-7MA4H8LUy8dbKmiyKynFtuvNt55_e2cse30bftRNM54OKhpZQIhpSeaCnAdFDOCR5gK5ryrCifFWMzSZAdx0eTf3B3UT&sig=AHIEtbTv69hjBdYNTZXQUWraukrH9OMagg

• (Absorption 1) x∧(x∨y) = x

(Absorption 2) x∨(x∧y) = x
http://en.wikipedia.org/wiki/Boolean_algebra

• Redundancy laws

The following laws will be proved with the basic laws. Counter-intuitively, it is sometimes necessary to complicate the formula before simplifying it.

Absorption

x + x · y = x
Proof:
x + x · y
= x · 1 + x · y
= x · (1 + y)
= x · 1
= x
x · (x + y) = x
Proof:
x · (x + y)
= (x + 0) · (x + y)
= x + (0 · y)
= x + 0
= x

http://nayuki.eigenstate.org/page/boolean-algebra-laws