I connettivi logici – Tutti?

Date due variabili logiche p e q le possibili funzioni in p e q sono 16

  p q   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
+-----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| 0 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 0 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
+-----+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+

Sono necessarie 16 funzioni logiche (connettivi logici)?


No, sono sufficienti NOT, AND, OR

+----+-----------------------------------------------
|  6 | p XOR q = NOT p AND q OR p AND NOT q
|  9 | p ⇔ q = p XNOR q = NOT p AND NOT q OR p AND q
| 13 | p ⇒ q = NOT(p AND NOT q) = NOT p OR q
| .. | ...
+----+-----------------------------------------------


No, sono sufficienti NOT, AND

+----+-----------------------------------------------
|  0 | 0
|  1 | p AND q
|  2 | p AND NOT q
|  3 | p
|  4 | NOT p AND q
|  5 | q
| 10 | NOT q
| 12 | NOT p
| 13 | NOT(p AND NOT q)
| 14 | NOT (p AND q)
| 15 | 1
| .. | ...
+----+-----------------------------------------------


No, sono sufficienti NOT, OR

+----+-----------------------------------------------
|  0 | 0
|  1 | NOT(NOT p OR NOT q)
|  2 | NOT(NOT p OR q))
|  3 | p
|  4 | p OR NOT q
|  5 | q
| 10 | NOT q
| 12 | NOT p
| 13 | NOT p OR q
| 14 | NOT p OR NOT q
| 15 | 1
| .. | ...
+----+-----------------------------------------------


No, è sufficiente NAND

+----+-----------------------------------------------
|  0 | 0
|  3 | p
|  5 | q
|  7 | (p NAND p) NAND (q NAND q)
| 10 | q NAND q
| 12 | p NAND p
| 14 | p NAND q
| 15 | 1
| .. | ...
+----+-----------------------------------------------


No, è sufficiente NOR

+----+-----------------------------------------------
|  0 | 0
|  1 | (p NOR p) NOR (q NOR q)
|  3 | p
|  5 | q
| 10 | q NOR q
| 12 | p NOR p
| 15 | 1
| .. | ...
+----+-----------------------------------------------