Mathematical Reasoning - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes
Chapter-14
MATHEMATICAL REASONING

---

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

For example: "A triangle has a four sides.", "New Delhi is the capital of India." are the statements.

2. Negaiton of a statement: Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.

3. Compound statement: 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.

The Compound statements are made by:
(i) Connectives: "AND", "OR"
(ii) Quantifiers: "There exists", "For every"
(iii) Implications: The meaning of implications “If ”, “only if ”, “ if and only if ”.

(a) "p ⇒$\Rightarrow$ q" : p is sufficient condition for q or p implies q.
q is necessary condition for p.
The converse of a statement p ⇒ q is the statement q ⇒ p.
p⇒ q together with its converse, gives p if and only if q.

(b) "p ⇔$\iff$ q"
A sentence with if p, then q can be written in the following ways.

• p implies q (denoted by p ⇒ q)
• p is a sufficient condition for q
• q is a necessary condition for p
• p only if q
• ∼q implies ∼p

4. Contrapositive: The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒∼p .

New statements, Special words/phrases

5. Contradiction : If to check whether p is true we assume negation p is true.

6. Validating statements: Checking of a statement whether it is true or false. The validity of a statement depends upon which of the special.

• The following methods are used to check the validity of statements:

(i) direct method

(ii) contrapositive method

(iii) method of contradiction

(iv) using a counter example.