## 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.

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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