Basic laws and properties of Boolean Algebra

Boolean Algebra

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In this tutorial we will learning about basic laws and properties of boolean algebra.

Boolean Algebra - Basic Postulates

Let X be a logical (binary) variable then,
if X is not 0
then, X is 1
and
if X is not 1
then, X is 0

OR relations (Logical Addition)
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1

AND relations (Logical Multiplication)
0 . 0 = 0
0 . 1 = 0
1 . 0 = 0
1 . 1 = 1

Complement Rule
~0 = 1
~1 = 0
where ~ is the NOT operator

Principle of Duality

Starting with a boolean relation, another boolean relation can be derived by
replacing each 0 with 1
replacing each 1 with 0
replacing each OR (+) with AND (.)
and replacing each AND (.) with OR (+)

Properties of 0

Let A be a logical (binary) variable
0 + A = A
0 . A = 0

Explanation

when A = 1
then, 0 + A = 0 + 1
= 1

and when A = 0
then, 0 + A = 0 + 0
= 0

so, output of 0 + A is equal to A

Similarly,
when A = 1
then, 0 . A = 0 . 1
= 0

and when A = 0
then, 0 . A = 0 . 0
= 0

so, output of 0 . A is always 0

Properties of 1

Let A be a logical (binary) variable
1 + A = 1
1 . A = A

Explanation

when A = 1
then, 1 + A = 1 + 1
= 1

and when A = 0
then, 1 + A = 1 + 0
= 1

so, output of 1 + A is always 1

Similarly,
when A = 1
then, 1 . A = 1 . 1
= 1

and when A = 0
then, 1 . A = 1 . 0
= 0

so, output of 1 . A is equal to A

Idempotence Law

Let A be a logical (binary) variable
A + A = A
A . A = A

Explanation

when A = 1
then, A + A = 1 + 1
= 1

and when A = 0
then, A + A = 0 + 0
= 0

Similarly,
when A = 1
then, A . A = 1 . 1
= 1

and when A = 0
then, A . A = 0 . 0
= 0

So, output of A + A and A . A is equal to A.

Involution

Let A be a logical (binary) variable
~(~A) = A

Explanation

when A = 1
then, ~A = 0
and ~(~A) = ~(0)
= 1

So, output of ~(~A) is A

Complementarity Law

Let A be a logical (binary) variable
A + (~A) = 1
A . (~A) = 0

Explanation

when A = 1
then, A + (~A) = 1 + 0
= 1

and when A = 0
then, A + (~A) = 0 + 1
= 1

so, output of A + (~A) is always 1

Similarly,
when A = 1
then, A . (~A) = 1 . 0
= 0

and when A = 0
then, A . (~A) = 0 . 1
= 0

So, output of A . (~A) is always 0

Commutative Law

Let A and B be two logical (binary) variables
A + B = B + A
A . B = B . A

Associative Law

Let A, B and C be three logical (binary) variables
(A + B) + C = A + (B + C)
(A . B) . C = A . (B . C)

Distributive Law

Let A, B and C be three logical (binary) variables
A + (B . C) = (A + B) . (A + C)
A . (B + C) = (A . B) + (A . C)

Absorption Law

Let A and B be two logical (binary) variables
A + (A . B) = A
A . (A + B) = A

Explanation

A + (A . B)
= A . (1 + B)
= A . 1
= A

Similarly,
A . (A + B)
= A.A + A.B
= A + A.B
= A . (1 + B)
= A . 1
= A

De Morgan's Law

Let A and B be two logical (binary) variables
~(A + B) = ~A . ~B
~(A . B) = ~A + ~B