Basic Theorems in Digital Electronics

Digital Electronics is a core and important subject of Electronics and Communication Engineering. Its simple and easy to understand. The theorems used in Digital electronics minimizes the switching function of digital circuits.

The main theorem is "De-Morgan's Theorem"

De-Morgan's Theorem states that i) Bubbled AND gate is same as Nor gate.
                                                          ii) Bubbled OR gate is same as NAND gate.

It implies that i) Ā + B̄ = Ā.B̄
                    ii) Ā.B̄ = Ā + B̄

Basic properties in Digital Electronics:               

1) Idempotent law : i) x + x = x
                              ii) x.x = x

2) Commutative Law : i) x + y = y + x
                                  ii) x.y = y.x

3) Associative law : i) x +(y + z)  = (x + y ) + z
                              ii) x.(y.z) = (x.y).z              

4) Complementary law : i) x + x̄ = 1
                                     ii) x.x̄ = 0

5) Distributive law : i) x(y + z) = x.y + y.z
                              ii) x + y.z = (x + y) + ( x + z)

6) Absorption law : i) x + xy = x
                              ii) x(x + y) = x
                             iii) x + x̄y = x + y
                             iv) x(x̄ + y) = xy

Consensum Theorem:

i) xy + x̄z + yz = xy + x̄z
ii) (x + y).(x̄ + z).(y + z) = (x + y).(x̄ + z)


All the above theorems and expressions are used to simplify a big equation in digital electronics.

Example: Simplify the expression : x̄.ȳ.z̄ + x̄.y.z̄ + x.ȳ.z̄ + x.y.z̄

Answer: above expression can be written as  x̄.z̄.(y + ȳ) + x.z̄.(y + ȳ)

it can be further written as  z̄.(x + x̄)

it can be again reduced as  z̄

So the big expression is reduced to z̄ using the above law's and theorems