Boolean Algebra | Engineering Mathematics - Civil Engineering (CE) PDF Download

Introduction

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

• Commutative Properties:
  (i) a+b = b+a
  (ii)a*b=b *a
• Distributive Properties
  (i) a+(b*c)=(a+b)*(a+c)
  (ii)a*(b+c)=(a*b)+(a*c)
• Identity Properties
  (i) a+0=a
  (ii) a *1=a
• Complemented Laws
  (i) a+a'=1
  (ii)a * a'=0

Sub-Algebra

Consider a Boolean-Algebra (B, *, +,', 0,1) and let A ⊆ B. Then (A,*, +,', 0,1) is called a sub-algebra or Sub-Boolean 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 sub-algebra 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 sub-algebra as it may not close the operation on B.

Isomorphic-Boolean Algebras

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.

1. a ≤b iff a+b=b
2. a ≤b iff a * b = a
3. Idempotent Laws
   (i)a+b=a
   (ii) a * a = a
4. Commutative Property
    (i) a+b=b+a
   (ii) a*b=b*a
5. Associative Property
   (i) a+(b+c)=(a+b)+c
   (ii) a*(b*c)=(a*b)*c
6. Absorption Laws
   (i) a+(a*b)=a
   (ii) a*(a+b)=a
7. Identity Laws
   (i) a+0=a  
   (ii) a*1=a
8. Null Laws
    (i) a*0=0
   (ii) a+1=1
9. Distributive Laws
   (i) a*(b+c)=(a*b)+(a*c)
   (ii) a+(b*c) = (a+b)*(a+c)   
10. Complement Laws
    (i)0'=1
    (ii) 1'=0
    (iii) a+a'=1
    (iv) a*a'=0
11. Involution Law
    (a')'=a
12. De Morgan's Laws
    (i) (a *b)'=(a' +b')
     (ii) (a+b)'=(a' *b')

Note:

1. 0 ≤ a ≤ 1 for every a ∈ B.
2. Every element b has a unique complement b'.

Boolean Functions

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 two-valued 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.