Question

What is a tautology in mathematical logic?

Answer 1

In mathematical logic, a tautology is a logical statement that is always true, regardless of the values of the variables in the statement. Tautologies are used to prove the validity of other statements by showing that they are logically equivalent to tautologies.

A tautology can be expressed using logical connectives such as "and" ( \(\wedge\), "or" ( \(\vee)\), "not" \((\neg)\), "implies" \((\rightarrow)\), and "if and only if" \((\leftrightarrow)\). For example, the statement "p \(\vee \neg p\) " is a tautology, because it is always true regardless of the value of \(p\). This is because the disjunction "p \(\vee \neg p\) " is true if either \(p\) is true or \(\neg p\) is true, and one of these must always be the case.

Tautologies can be used to prove the validity of other statements by showing that they are logically equivalent to tautologies.