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, \(the\ shape\ of\ the\ earth\ is\ flat\ and\ round\).
| \[\mathbf{p}\] | \[\mathbf{\sim p}\] | \[\mathbf{p \land \sim p}\] |
|---|---|---|
| \[T\] | \[F\] | \[F\] |
| \[F\] | \[T\] | \[F\] |