Javatpoint Logo
Javatpoint Logo

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
      (ii)a*b=b *a

2. Distributive Properties

      (i) a+(b*c)=(a+b)*(a+c)
      (ii)a*(b+c)=(a*b)+(a*c)

3. Identity Properties

      (i) a+0=a
      (ii) a *1=a

4. 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:

Boolean Algebra

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

(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

Note: A function can always be described in tabular form. An alternative way of expressing the functions is specifying the function by an expression.


Next TopicBoolean Expression




Help Others, Please Share

facebook twitter google plus pinterest

Learn Latest Tutorials


Preparation


Trending Technologies


B.Tech / MCA