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A tautology is a formula whose negation is unsatisfiable. Roughly spoken, a tautoloy is always true. For example,
This statement is either true or false.
A natural number is either even or odd.
There is a special symbol that denotes a tautology. The symbol represents a statement that is a tautology.
With it, we can write that any statement is a tautology by simply writing:
Law of the excluded middle
The Law of the excluded middle states that either a statement is true or the statement's negation is true.
Let be a statement, then we can also reword the Law of the excluded middle as
where denotes the exclusive or operation.
Reductio ad absurdum
De Morgan's Law
¬(p ∧ q) ⇔ ¬p ∨ ¬q
¬(p ∨ q) ⇔ ¬p ∧ ¬q
Proof by cases
If there are variables in a formula, we check the possible valuations of the formula in a truth table