Tautology and contradiction - SS2 Mathematics Lesson Note

A compound statement that is always true irrespective of the truth values of its sub-statements is a tautology. On the reverse, a compound statement whose truth value is always false no matter what the truth values of its sub-statements are is a contradiction.

The statement \(p \vee \sim p\) ( \(p\ or\ not\ p\)) is a tautology. Example, \(our\ second\ child\ is\ either\ a\ male\ or\ female.\)

\[\mathbf{p}\]	\[\mathbf{\sim p}\]	\[\mathbf{p \vee \sim p}\]
\[T\]	\[F\]	\[T\]
\[F\]	\[T\]	\[T\]

The statement \(p \land \sim p\) ( \(p\ and\ not\ p\)) is a contradiction. Example, \(the\ shape\ of\ the\ earth\ is\ flat\ and\ round\).

\[\mathbf{p}\]	\[\mathbf{\sim p}\]	\[\mathbf{p \land \sim p}\]
\[T\]	\[F\]	\[F\]
\[F\]	\[T\]	\[F\]