The functionally complete set is by which you can perform all operations. So, if any logical set is able to implement the operation {And , NOT} or {OR, NOT}; it is known as functionally complete.
Now come to NOR gate.
- $A$(NOR)$B= (A+B)'$
- $A$(NOR)$A =(A+A)' =(A)'$ , so we can perform the NOT operation .
- $(A+B)'$ NOR $(A+B)' =((A+B)' + (A+B)')'=((A+B)')' =(A+B) $, so OR operation is also performed successfully .
So, NOR is functionally complete.