Universal Property of NAND and NOR Gates

Universal property :

NAND and NOR gates possess a special property: they are universal. That is, given enough gates, either type of gate is able to mimic the operation of any other gate type. For example, it is possible to build a circuit exhibiting the OR function using three interconnected NAND gates as shown in below. The ability for a single gate type to be able to mimic any other gate type is one enjoyed only by the NAND and the NOR. In fact, digital control systems have been designed around nothing but either NAND or NOR gates, all the necessary logic functions being derived from collections of interconnected NANDs or NORs.

                                                            step1                         step2   
As shown in above figure 
step 1 :
              A . A = A (Idempotent law)
             complement (A.A ) = complement (A)
step2 : 
                          complement ( complement(A) ) = A (double negation law)
                          complement (A . B) = complement (A) + complement (B) (deMorgan's law)

complement ( complement(A) . complement(B) ) = complement ( complement(A) ) + complement (complement (B))
                                                                              =  A + B (OR of A,B)



Universal property of NAND Gates :

Universal property of NOR Gates :

Laws followed in digital electronics

Laws followed by digital electronics :

Annulment Law :
if A (is unknown value, means whether 0 or 1) is AND'ed with 0 then result always equal to  0.
if A is OR'ed with 1 then the result always equal to 1.
                      A . 0 = 0
                      A + 1 = 1

Identity Law :
The result always equal to  A when  A is OR'ed with 0 or if A is AND'ed with 1.
                      A . 1 = A
                       A + 0 = A

Idempotent Law :
The result is always equal to A when OR'ed or AND'ed with  A itself.
                       A + A = A
                       A . A = A

Complement Law :
The result is always equal to 0 when AND'ed with complement of itself.
The result is always equal to 1 when OR'ed with complement of itself.
                       A . ABAR = 0
                       A + ABAR = 1

Commutative Law :
 The order of application of two separate  inputs is not important.
                         A + B = B+ A

Double Negation Law :

if A is two times complemented it gives same value.
                        complement (complement ( A) ) = A

de Morgan's Law:

Law1 : NOR'ed of two inputs is equal to  AND'ed of complement inputs.
Law2 : NAND'ed of two inputs is equal to OR'ed of complement inputs.
             complement (A + B) = complement(A) . complement(B)
             complement (A . B ) = complement(A) + complement(B)