# Mathematical Statements

Mathematical Statement:

A meaningful composition of words which can be considered either true or false is called a mathematical statement or simply a Statement.

A single letter shall be used to denote a statement. For example, letter ‘p’ may be used to stand for the statement “ABC is an equilateral triangle.” Thus, p = ABC is an equilateral triangle.

Truth Value of a Statement:

A statement is said to have truth value T or Faccording as the statement considered is true or false. For example, the statement ‘2 plus 2 is four’ has truth value T, whereas, the statement ‘2 plus 2 is five’ has truth value F. The knowledge of truth value of statements enables to replace one statement by some other (equivalent) statement(s).

Production of New Statement:

New statements from given statements can be produced by

(i) Negation: $\sim$
If p is a statement then its negation ‘$\sim$p’ is statement ‘not p’. ‘$\sim$p’ has truth value F or T according as the truth value of  ‘p’ is T or F.

(ii) Implication: $\Rightarrow$
If from a statement p another statement q follows, we say ‘p implies q’ and write ‘p$\Rightarrow$ q’. Such a result is called an implication. The truth value of ‘p$\Rightarrow$q’ is F only when p has truth value T and q has the truth value F.
The statements involving ‘if p holds then q’ are of the kind p$\Rightarrow$q. For example, $x = 2 \Rightarrow {x^2} = 4$.

(iii) Conjunction: $\wedge$
The sentence ‘p and q’ which may be denoted by ‘p$\wedge$q’ is the conjunction of p and q. Truth value of p$\wedge$q is T only when both p and q are true.

(iv) Disjunction: $\vee$
The sentence ‘p and q (or both)’ which may be denoted by ‘p$\vee$q’ is called the disjunction of the statements p and q. Truth value of p$\vee$q is F only when both p and q are false.

Equivalence of Two Statements, p$\Leftrightarrow$q:

Two statements p and q are said to be equivalent if one implies the other and in such a case using the double implication symbol,$\Leftrightarrow$, we write p$\Leftrightarrow$q.

The statements which involve the phrase ‘if and only if’ or ‘is equivalent to’ or “the necessary and sufficient conditions” are of the kind p$\Leftrightarrow$q. For example, ABC is an equilateral triangle AB = BC = CA.

For brevity, the phrase ‘if and only if’ is shortened to “iff”. As described above, the symbols $\vee$ stand for the words ‘and’, ‘or’ respectively. Disjunction symbol $\vee$ is used in the logical sense ‘and/or’. The symbols$\wedge$, $\vee$and are logical connectives and are frequently used.

Following is the table showing truth values of different compositions of statements. Such tables are called truth tables.

 p q $\sim$p $\sim$q p$\Rightarrow$q p$\wedge$q p$\vee$q p$\Leftrightarrow$q T T F F T T T T T F F T F F T F F T T F T F T F F F T T T F F T

By forming truth tables equivalence of various statements can easily be ascertained. For example, we shall easily see that the implication ‘p$\Rightarrow$q’ is equivalent to ‘$\sim$p$\Rightarrow$$\sim$q’. The implication ‘$\sim$q$\Rightarrow$$\sim$p’ is called contra positive of p$\Rightarrow$q.