## CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning

**Statements**

A statement is a sentence which is either true or false, but not both simultaneously.

Note:

No sentence can be called a statement if

- It is an exclamation.
- It is an order or request.
- It is a question.

**Simple Statements**

A statement is called simple if it cannot be broken down into two or more statements.

**Compound Statements**

A compound statement is the one which is made up of two or more simple statement.

**Connectives**

The words which combine or change simple statements to form new statements or compound statements are called connectives.

**Conjunction**

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q is written in symbolic form as “p ∧ q”.

Note:

- The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T.
- The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.

**Disjunction**

If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.

Note:

- The statement p ∨ q has the truth value F whenever both p and q have the truth value F.
- The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.

Negation

An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement p in symbolic form is written as “~p”.

Note:

- ~p has truth value T whenever p has truth value F.
- ~p has truth value F whenever p has truth value T.

**Negation of Conjunction**

The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q.

Equivalently we write ~ (p ∧ q) = ~p ∨ ~q.

**Negation of Disjunction**

The negation of a disjunction p v q is the conjunction of negation of p and the negation of q.

Equivalently, we write ~(p ∨ q) = ~p ∧ ~q.

**Negation of Negation**

Negation of negation of a statement is the statement itself.

Equivalently, we write ~(~p) = p

**The Conditional Statement**

If p and q are any two statements, then the compound statement “if p then g” formed by joining p and q by a connective ‘if-then’ is called a conditional statement or an implication and is written in symbolically p → q or p ⇒ q, here p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q).

**Contrapositive of Conditional Statement**

The statement “(~q) → (~p) ” is called the contrapositive of the statement p → q.

**Converse of a Conditional Statement**

The conditional statement “q → p” is called the converse of the conditional statement “p → q”.

**Inverse of Conditional Statement**

The Conditional statement “q → p” is called inverse of p → q.

**The Biconditional Statement**

If two statements p and q are connected by the connective ‘if and only if’, then the resulting compound statement “p if and only if q” is called biconditional of p and q and is written in symbolic form as p ⇔ q.

**Quantifier**

(i) For all or for every is called universal quantifier.

(ii) There exists is called existential quantifier.

**Validity of Statements**

A statement is said to valid or invalid according to as it is true or false.

If p and q are two mathematical statements, then the statement

(i) “p and q” is true if both p and q are true.

(ii) “p or g” is true if p is false

⇒ q is true orq is false ⇒ p is true.

(iii) “If p, then q” is true p is true ⇒ q is true

or

q is false

⇒ p is false

or

p is true and q is false less us to a contradiction,

(iv) “p if and only if q” is true, if

(a) p is true ⇒ q is true and

(b) q is true ⇒ p is true.