Question

Determine whether the following propositions is a Tautology or a contradiction or neither.

(i) (p ∧ q) ∧ ~p

(ii) [~p ∧ (p ∨ q)]

(iii) (p ∧ q) → (p ∨ q)

(iv) (p ∧ q) →p

(v) ~ p ∧ ~q

Answer 1

(i) (p ∧ q) ∧~p

| p | q | p ∧ q | ~p | (p ∧ q) ∧~p |
| T | T | T | F | F |
| T | F | F | F | F |
| F | T | F | T | F |
| F | F | F | T | F |

From last column we conclude that it is a contradiction

(ii) [~p ∧ (p ∨ q)]

| p | ~p | q | p ∨ q | ~p ∧ (p ∨ q) |
| T | F | T | T | F |
| T | F | F | T | F |
| F | T | T | T | T |
| F | T | F | F | F |

From last column we conclude it is neither tautology nor a contradiction

(iii) (p ∧ q) → (p ∨ q)

| p | q | p ∧ q | p ∨ q | (p ∧ q) → (p ∨ q) |
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | T |

From last column we conclude it is a tautology

(iv) (p ∧ q) → p

| p | q | p ∧ q | (p ∧ q) → p |
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |

From last column we conclude that it is a tautology

(v) ~ p ∧ ~q

| p | q | ~ p | ~q | ~ p ∧ ~q |
| T | T | F | F | F |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | T |

It is neither tautology nor contradiction