| Name: | Symbol: | Usage: | Meaning of example: | Some equivalences: |
| Exclusive OR | ![]() | p q | "Either p or q, but not both, is true." | (p \/ q) /\ ¬(p /\ q) |
| Implies |
| p q
| "If p is true, then q is also true." | ¬p \/ q ¬(p /\ ¬q) ¬q ¬p
|
| Biconditional |
| p q
| "p is true if and only if q is true." | (p q) /\ (q p)¬(p q)
|
The truth tables for these operators are as follows:
| Cases | Truth values of... | |||||
| # | p | q | p q
| p q
| p q
| |
| 0 | F | F | F | T | T | |
| 1 | F | T | T | T | F | |
| 2 | T | F | T | F | F | |
| 3 | T | T | F | T | T | |