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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.
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 D70 whose Hasse diagram is shown in fig:
Clearly, A= {1, 7, 10, 70} and B = {1, 2, 35, 70} is a subalgebra of D70. 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.
Two Boolean algebras B and B1 are called isomorphic if there is a one to one correspondence f: B⟶B1 which preserves the three operations +,* and ' for any elements a, b in B i.e.,
f (a+b)=f(a)+f(b)
f (a*b)=f(a)*f(b) and f(a')=f(a)'.
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.
Note:
 0 ≤ a ≤ 1 for every a ∈ B.
 Every element b has a unique complement b'.
Consider the Boolean algebra (B, ∨,∧,',0,1). A function from A''to A is called a Boolean Function if a Boolean Expression of n variables can specify it.
For the twovalued Boolean algebra, any function from [0, 1]n to [0, 1] is a Boolean function.
Example1: The table shows a function f from {0, 1}3 to {0, 1}
Example2: The table shows a function f from {0, 1, 2, 3}2 to {0,1,2,3}.
Note: A function can always be described in tabular form. An alternative way of expressing the functions is specifying the function by an expression.
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47 videos119 docs75 tests
