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 = 0 + 0 = 0

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