JEE Main & Advanced Mathematics Mathematical Logic and Boolean Algebra Statements or Propositions

Propositions : A statement or a proposition is an assertive (or declarative) sentence which is either true or false but not both a true statement is called valid statement. If a statement is false, then it is called invalid statement.

Open statement : A declarative sentence containing variable (s) is an open statement if it becomes a statement when the variable (s) is (are) replaced by some definite value (s).

Truth Set : The set of all those values of the variable (s) in an open statement for which it becomes a true statement is called the truth set of the open statement.

Truth Value : The truth or falsity of a statement is called its truth value.

If a statement is true, then we say that its truth value is ‘True’ or ‘T’. On the other hand the truth value of a false statement is ‘False’ or ‘F’.

Logical variables : In the study of logic, statements are represented by lower case letters such as p, q, r, s.. These letters are called logical variables.

For example, the statement ‘The sun is a star’ may be represented or denoted by p and we write p : The sun is a star  

Similarly, we may denote the statement \[145=\text{ }2\].

Quantifiers : The symbol \[\forall \](stands for ‘for all’) and \[\exists \](stands for “there exists”) are known as quantifiers.

In other word, quantifiers are symbols used to denote a group of words or a phrase.

The symbols \[\forall \] and \[\exists \] are known as existential quantifiers. An open sentence used with quantifiers always becomes a statement.

Quantified statements : The statements containing quantifiers are known as quantified statements.

\[{{x}^{2}}>0.\forall x\in R\] is a quantified statement. Its truth value is T.