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. Properties of Boolean Algebra:1. Commutative Properties: (i)a+b = b+a 2. Distributive Properties (i) a+(b*c)=(a+b)*(a+c) 3. Identity Properties (i) a+0=a 4. Complemented Laws: (i) a+a'=1 SubAlgebra:Consider a BooleanAlgebra (B, *, +,', 0,1) and let A ⊆ B. Then (A,*, +,', 0,1) is called a subalgebra or SubBoolean Algebra of B if A itself is a Boolean Algebra i.e., A contains the elements 0 and 1 and is closed under the operations *, + and '. Example: Consider the Boolean algebra D_{70} whose Hasse diagram is shown in fig: Clearly, A= {1, 7, 10, 70} and B = {1, 2, 35, 70} is a subalgebra of D_{70}. Since both A and B are closed under operation ∧,∨and '. Note: A subset of a Boolean Algebra can be a Boolean algebra, but it may or may not be subalgebra as it may not close the operation on B.IsomorphicBoolean Algebras:Two Boolean algebras B and B_{1} are called isomorphic if there is a one to one correspondence f: B⟶B_{1} which preserves the three operations +,* and ' for any elements a, b in B i.e., Example: The following are two distinct Boolean algebras with two elements which are isomorphic. 1.The first one is a Boolean Algebra that is derived from a power set P(S) under ⊆ (set inclusion),i.e., let S = {a}, then B = {P(S), ∪,∩,'} is a Boolean algebra with two elements P(S) = {∅,{a}}. 2. The second one is a Boolean algebra {B, ∨,∧,'} with two elements 1 and p {here p is a prime number} under operation divides i.e., let B = {1, p}. So, we have 1 ∧ p = 1 and 1 ∨ p = p also 1'=p and p'=1. The table shows all the basic properties of a Boolean algebra (B, *, +, ', 0, 1) for any elements a, b, c belongs to B. The greatest and least elements of B are denoted by 1 and 0 respectively. 1. a ≤b iff a+b=b 2. a ≤b iff a * b = a Note:

(x, y, z)  f 
(0, 0, 0)  0 
(0, 0, 1)  0 
(0, 1, 0)  1 
(0, 1, 1)  0 
(1, 0, 0)  1 
(1, 0, 1)  1 
(1, 1, 0)  0 
(1, 1, 1)  1 
Example2: The table shows a function f from {0, 1, 2, 3}^{2} to {0,1,2,3}.
(x, y)  f 
(0, 0)  1 
(0, 1)  0 
(0, 2)  0 
(0, 3)  3 
(1, 0)  1 
(1, 1)  1 
(1, 2)  0 
(1, 3)  3 
(2, 0)  2 
(2, 1)  0 
(2, 2)  1 
(2, 3)  1 
(3, 0)  3 
(3, 1)  0 
(3, 2)  0 
(3, 3)  2 