A mathematically acceptable statement is a sentence which is either true or false but not both. For example,
- Two plus two equals four.
- The sum of two positive numbers is positive.
- All prime numbers are odd numbers.
Of these sentences, the first two are true and the third one is false.
Negation of a Statement
The denial of a statement is called the negation of the statement. If p denote a statement, then the negation of p is denoted by ∼p, and read as ‘not p’.
A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement.
Some of the connecting words which are found in compound statements like “And”, “Or”, etc. are often used in Mathematical Statements. These are called connectives.
The compound statement with ‘And’ is true if all its component statements are true.
A compound statement with an ‘Or’ is true when one component statement is true or both the component statements are true.
The contrapositive of a statement p ⇒ q is the statement ∼q ⇒ ∼p
The converse of a statement p ⇒ q is the statement q ⇒ p
Validity of Statements
The following methods are used to check the validity of statements:
- direct method
- contrapositive method
- method of contradiction
- using a counter example