JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Tautologies and Contradictions

Let \[p,q,r,....\] be statements, then any statement involving \[p,q,r\],....and the logical connectives \[\wedge ,\vee ,\tilde{\ },\Rightarrow ,\Leftrightarrow \] is called a statement pattern or a Well Formed Formula (WFF).

For example

(i) \[p\,\vee \,q\]

(ii)  \[p\Rightarrow q\]

(iii) \[((p\wedge q)\vee r)\Rightarrow (s\wedge \tilde{\ }s)\]

(iv) \[(p\Rightarrow q)\Leftrightarrow (\tilde{\ }q\Rightarrow \tilde{\ }p)\]etc.

are statement patterns.

A statement is also a statement pattern.

Thus, we can define statement pattern as follows.

Statement pattern : A compound statement with the repetitive use of the logical connectives is called a statement pattern or a well- formed formula.

Tautology : A statement pattern is called a tautology, if it is always true, whatever may be the truth values of constitute statements.

A tautology is called a theorem or a logically valid statement pattern. A tautology, contains only T in the last column of its truth table.

Contradiction : A statement pattern is called a contradiction, if it is always false, whatever may the truth values of its constitute statements.

In the last column of the truth table of contradiction there is always F.

• The negation of a tautology is a contradiction and vice versa.