Given the amount of terminology and symbology being used, I thought it might be a good idea to summarize all the possible operations for a two-input gate. Of course these range from the trivial to the regular to the obscure, but we'll see...
| Category | Truth | Description | Symbol | Gate |
|---|---|---|---|---|
| 0 on | 00|0 | Always low | 0 | |
| 1 on | 00|1 | NOR | ↓ ∨̅ |  |
00|0 | Abjunction (NOT A AND B) | ↛ |  | |
00|0 | Abjunction (A AND NOT B) | ↛ |  | |
00|0 | AND | & ∧ |  | |
| 2 on | 00|1 | NOT A | A̅ | |
00|1 | NOT B | B̅ | ||
00|1 | Equivalence, XNOR | A = B |  | |
00|0 | Exclusive OR, XOR | ⊕ |  | |
00|0 | B | B | ||
00|0 | A | A | ||
| 3 on | 00|1 | NAND | &̅ ↑ ∧̅ |  |
00|1 | Material conditional (negation of abjunction) NOT A OR B | → |  | |
00|1 | Material conditional (negation of abjunction) A OR NOT B | → |  | |
00|0 | OR | ∨ | |  | |
| 4 on | 00|1 | Always high | 1 | |