Equivalent Statements, Tautology, and Contradiction
Logic is the art of going wrong with confidence.
—Morris Kline (1908–1992)

 

Definition.  Two propositions (statements) are equivalent if and only if they have the same truth values (in all cases).

Example.  Show that  and  are equivalent statements.
It is not true that an angle is acute or obtuse. An angle is not acute and not obtuse.

      The two statements are equivalent, since the truth values are the same; note the last column in each table are the same. We have shown one of DeMorgan's Laws; the other of DeMorgan's Laws is that  and  are equivalent statements.

Definition. The denial of a statement is a statement that is equivalent to the negation of the statement.

Example. The denial of "The parallelogram is a square."  is "The parallelogram is not a square."

Definition. A tautology is a statement that is true in all cases.

Example.   is a tautology. 
      The lines l and m are parallel or the lines l and m are not parallel.

Definition. A contradiction is a statement that is false in all cases.

Example.   is a contradiction.
      The lines l and m are parallel and the lines l and m are not parallel.