A truth table sets out all possible truth value combinations for the simple component statements and shows each value of the compont statement.
AND
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
OR (Inclusive)
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
OR (Exclusive)
| p | q | p ⊻ q |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
IMPLICATION
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
IFF
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Tautologies and Contradictions
Tautologies and contradictions can be used to construct arguments and proofs.
| p | ¬p | p ∨ ¬p |
|---|---|---|
| T | F | T |
| F | T | T |
p ∨ ¬p is an example of a Tautology. p ∧ ¬p is an example of a contradiction since it cannot be true (whatever value p assumes).