Note. You can use /\ for ∧, \/ for ∨ and -> for → to simplify typing answers.
There are different styles of truth tables. Here, I show a style where full compound propositions are shown.
¬(p → q) → p
| p | q | p → q | ¬(p → q) | ¬(p → q) → p |
|---|---|---|---|---|
| F | F | T | F | T |
| F | T | T | F | T |
| T | F | F | T | T |
| T | T | T | F | T |
¬(p ∨ q) ↔ ¬p ∧ ¬q
| p | q | p ∨ q | ¬(p ∨ q) | ¬p | ¬q | ¬p ∧ ¬q | ¬(p ∨ q) ↔ ¬p ∧ ¬q |
|---|---|---|---|---|---|---|---|
| F | F | F | T | T | T | T | T |
| F | T | T | F | T | F | F | T |
| T | F | T | F | F | T | F | T |
| T | T | T | F | F | F | F | T |
(¬p ∧ (p ∨ q)) → q
| p | q | ¬p | p ∨ q | ¬p ∧ (p ∨ q) | (¬p ∧ (p ∨ q)) → q |
|---|---|---|---|---|---|
| F | F | T | F | F | T |
| F | T | T | T | T | T |
| T | F | F | T | F | T |
| T | T | F | T | F | T |
((p ∧ q) → r) → ((p ∧ ¬r) → ¬q)
| p | q | r | p ∧ q | (p ∧ q) → r | ¬r | p ∧ ¬r | ¬q | (p ∧ ¬r) → ¬q | ((p ∧ q) → r) → ((p ∧ ¬r) → ¬q) |
|---|---|---|---|---|---|---|---|---|---|
| F | F | F | F | T | T | F | T | T | T |
| F | F | T | F | T | F | F | T | T | T |
| F | T | F | F | T | T | F | F | T | T |
| F | T | T | F | T | F | F | F | T | T |
| T | F | F | F | T | T | T | T | T | T |
| T | F | T | F | T | F | F | T | T | T |
| T | T | F | T | F | T | T | F | F | T |
| T | T | T | T | T | F | F | F | T | T |
p ∨ ¬q
¬(¬p ∨ a) ∨ r
or(p ∧ ¬a) ∨ r
(by DeMorgan's Law)
¬(p ∨ q) ∨ ¬p
or(¬p ∧ ¬q) ∨ ¬p
(by DeMorgan's Law) or¬p
(by the Absorption Law).