Statements
* A sentence is called a mathematically acceptable statement if it is either true or false.
* A statement is called simple if it cannot be broken into two or more statement.
* The denial of a statement is called the negation of the statement.
* If p is a statement, then the negation of p is also a statement, and is denoted by ~p.
This is read as ‘not p’.
* A compound statement is obtained by combining one or more simple statements
using some connecting words like “and”, “or”, etc.
* The simple statements that form a compound statement are called component statements.
Compound Statements
* A compound statement with the connective word “and,” is called a conjugation.
* The truth value of a conjugation is true, if the truth values of all the component statements are truth.
* The conjugation is false, if any one of the component statements is false.
* A compound statement with the connective “or” is called a disjunction.
* A compound statement with the connective “or” is true, when at least one component statement is true.
* A compound statement with “or” is false, when all its component statements are false.
* The quantifiers “for all”, “for every” and “for no” are called universal quantifiers.
The universal quantifier is denoted by ∀.
* The quantifiers “there exists”, “for some” are called existential quantifiers. The
existential quantifiers is denoted by ∃.
Implications
* A compound statement in the form, “if p, then q”, is called an implication or a conditional statement.
“If p, then q” is represented as p ⇒ q.
The converse of p ⇒ q is q ⇒p.
The inverse of p ⇒q is ~p ⇒~q.
The contrapositive of p ⇒q is ~q ⇒~p.
* A compound statement involving statements p and q which is in the form of “p if and only if q”, is called a bi-implication or a bi-conditional.
“p if and only if q” is represented as p ⇔ q.
Validating Statements
* To validate where a conjugation is true, we have to show that all of its component statements are true.
* To validate whether a disjunction is true, we have to show that one of its component statements is true.
* To prove the statement “if p then q”, we need to show that any one of the following case is true.
* Case 1: By assuming that p is true, show that q must be true. (Direct
Method)
* Case 2: By assuming that q is false, show that p must be false.
(Contrapositive Method)
* To prove the statement “p if and only if q”, we need to show the following:
* If p is true, then q is true.
* If q is true, then p is true.
Contradiction Method: To prove that a statement p is true, first, we assume that p is
not true or negation p is true. Then, we arrive at some result that contradicts the
assumptions. This concludes that p is true.
Counter Example: The example of a situation where a statement is not valid.