Tautologies
A proposition P is a tautology if it is true under all circumstances. It means it contains the only T in the final column of its truth table.
Example: Prove that the statement (p⟶q) ↔(∼q⟶∼p) is a tautology.
Solution: Make the truth table of the above statement:
| p | q | p→q | ~q | ~p | ~q⟶∼p | (p→q)⟷( ~q⟶~p) |
| T | T | T | F | F | T | T |
| T | F | F | T | F | F | T |
| F | T | T | F | T | T | T |
| F | F | T | T | T | T | T |
As the final column contains all T’s, so it is a tautology.
Contradiction:
A statement that is always false is known as a contradiction.
Example: Show that the statement p ∧∼p is a contradiction.
Solution:
| p | ∼p | p ∧∼p |
| T | F | F |
| F | T | F |
Since, the last column contains all F’s, so it’s a contradiction.
Contingency:
A statement that can be either true or false depending on the truth values of its variables is called a contingency.
| p | q | p →q | p∧q | (p →q)⟶ (p∧q ) |
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | F |
| F | F | T | F | F |
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