By going through these CBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning Class 11 Notes, students can recall all the concepts quickly.

## Mathematical Reasoning Notes Class 11 Maths Chapter 14

Statement : A sentence is called mathematically acceptable statement, if it is either true of false but not both.

Negation of a Statement: The denial of a statement is called negation of the statement e.g.

If p : Diagonals of a rectangle are equal.

So, ~ p : Diagonals of a rectangle are not equal. This may also be written as

~ p : It is false that diagonals of a rectangle are equal. Further, it may also be written as

~ p : There is at least one rectangle whose diagonals are not equal.

Compound Statement: A compound statement is a statement which is made up of two or more statements.

Each statement is called a component statement.

The Connecting Word “And” : We can connect the two statements by the word “AND”, e.g. ‘

p : 48 is divisible by 4.

q : 48 is divisible by 6.

p and q : 48 is divisible by 4 and 6.

Truth Value of p and q : The statementp and q are both true, otherwise it is false, i.e., it is false, when

(i) p is true and q is false.

(ii) p is false and q is true.

(iii) p is false and q is false.

The Connecting Word “OR” : The statement p and q may be connected by the connecting word ‘OR’: i.e., p or q, e.g.

p : Ice cream is available at dinner.

q : Coffee is available at dinner.

p or q : Ice cream or coffee is available at dinner.

Exclusive “OR” : In a statement p or q, if one of the statements either p or q is true, then the statement is true. Thus, connecting of word “OR” is exclusive.

Inclusive “OR” : In a statement either both are true, then connecting word ‘OR’ is inclusive, e.g.

At plus one level, a student can either opt for Biology or Mathematics or both.

Here, the connecting word “OR” is inclusive.

Truth value of p or q : When p and q statements both are false, then p or q is also false, otherwise it is true.

Thus, p or q is true, when

(i) p is true, q is false.

(ii) p is false, q is true.

(iii.) p and q both are true.

Quantifier “There Exists” : There exists, is used for at least one e.g.

p : There exist, a quadrilateral whose all sides are equal.

The statement is equivalent to

There is at least one quadrilateral whose all sides are equal.

Quantifier “For All” : In mathematical statements “for all” is commonly used e.g. For all n ∈ N implies that each n is a natural rumber.

Implications : “If then”, “only if’ and “if and only if’ are known as imphcations. .

If p then q : The statement if “p then q” says that in the event if p is true, the i q must be true e.g.

‘ I a numbei ” a multiple of 4, then it is a multiple of 2.

Here p : A ni her is a multiple of 4.

and q : The m. aber is a multiple of 2.

When p is true i.e., a number is a multiple of 4, then q is true

i. e., the number is a multiple of 2.

“If p then q” may be used as : l.p implies q : It is denoted by p ⇒ q. The symbol ⇒ stands for implies.

2. p is sufficient condition for q : A number is a multiple of 4 is sufficient to conclude that the number is a multiple of

3. p only if q. The number is a multiple of 4 only if it is a multiple of 2.

4. q is a necessary condition for p. When the number is a multiple of 4, it is necessarily a multiple of 2.

5. – q implies ~ p (~ stands for negation or not)

Converse Statement : The converse of “if p then q” is if q then p e.g.

If a number x is odd, then x^{2} is also odd.

Its converse is : If x^{2} is odd, then x is also odd.

Contrapositive Statement: Contrapositive of “if p then q” is if not q then not p. e.g. contrapositive of

“If a number is divisible by 4, it is divisible by 2”, is

“If a number is not divisible by 2, then it is not divisible by 4”.

Truth Value of “If p then q” : Truth value of the statement “if p then q” is false, when p is true and q is false, otherwise it is true, ue., It is true, when

(i) p is true, q is true.

(it) p is false, q is true.

(iii) p is false, q is false.

However, three methods are adopted to test the truth value of this statement.

(1) Assuming that p is true, prove that q must be true. (Direct method).

(2) Assuming that q is false, prove thatp must be false.

(3) Assuming thatp is true and q is false, obtain a contradiction. (Contradiction method).

Statements with “If and Only If’: “If and only if’, represented by the symbol ⇔, has the following equivalent forms :

(i) p if and only if q.

(it) q if and only p.

(iii) p is necessary and sufficient condition for q and vice-verse.

(iv) p ⇔ q.

Truth Value of “If and only If” : Truth value of statements with “if and only iP is true when p and q are both true or false, otherwise it is false, ue.., statement if and only if is true, when

(i) p is true, q is true.

(ii) p is false, q is false.

Statement with if and one if is false, when

(i) p is true, q is false

(ii) p is false, q is true.

To, Prove A Statement By Contradiction

Example : To prove that √3 is irrational.

Let √3 be rational. So, √3 = p/q, where p and q are co-prime.

p^{2} = 3q^{2} ⇒ 3 divides p or 3 is a factor of p.

Let p = 3k. ∴p^{2} = 9k^{2}.

∴ p^{2} = 3q^{2}becomes 9k^{2} = 3q^{2} or 3k^{2} = q^{2}.

This means 3 is a factor or q.

i. e., 3 is a factor of p and q both. This is a contradiction.

∴ √3 is irrational.

Counter Example : To prove that a statement is false, we find an example (called counter example) by which the given statement is not valid. Then, we say that the statement is false.