Mathematics__Boolean__Algebra__Class__Notes

RELATIONSHIP BETWEEN BOOLEAN ALGEBRA AND LATTICE
(Boolean algebra as lattice)

A lattice L is a partially ordered set in which every pair of elements x, y ∈ L
has a least upper bound denoted by l u b (x, y) and a greatest lower bound
denoted by g l b (x, y).

The two operations of meet and join denoted by and∧∨ respectively defined
for any pair of elementsx, y ∈ L as




A lattice L with two operations of meet and join shall be a Boolean algbera if
L is

1. Complemented: i.e.

(i) If must have a least element 0 and a greatest element 1 and
(ii) For every element x ∈ L these must exist an element x′∈ L such that




2. Distributed:




Boolean Algebra:

A complemented distributive lattice is known as a Boolean Algebra. It is
denoted by (B, ∧,∨,',0,1), where B is a set on which two binary operations ∧
(*) and ∨(+) and a unary operation (complement) are defined. Here 0 and 1
are two distinct elements of B.

Since (B,∧,∨) is a complemented distributive lattice, therefore each element
of B has a unique complement.

, Ex: (P(A), *, +, ‘, 0,1) where A ={1,2} and P(A) – Power set of A –Boolean
Algebra


Ex: S6 is boolean algebra or not
S6={1,2,3,6}
Element Complement GLB (0) - 1 LUB (1) - 6
1 6 1 6
2 3 1 6
3 2 1 6
6 1 1 6


From above table we can observe that all elements have only one
complement. So S6 is complemented and distributed lattice so it is also
called boolean algebra.