RBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning

These comprehensive RBSE Class 11 Maths Notes Chapter 14 Mathematical Reasoning will give a brief overview of all the concepts.

RBSE Class 11 Maths Chapter 14 Notes Mathematical Reasoning

Introduction:
It is used to say that over the centuries human race grow from its lowest level to highest level in comparison with all the species available. Its main reason was that it has analystical and analysis capacity.
In this lesson, we shall learn about some basic ideas of mathematical reasoning and the process of reasoning especially in context of mathematics. In mathematical language, there are two kinds of reasoning :
(i) Inductive Reasoning
(ii) Deductive Reasoning
We have already discussed the inductive reasoning in Mathematical induction. Now, we shall discuss some fundamentals of deductive reasoning.

Statements
The basic unit involved in mathematical reasoning is a mathematical statement “ A sentence is called a mathematically acceptable statement if it is either true or false but not both at the same time.”
True statement is called valid statement and false statement is called invalid statement.
For example :
(i) Pt. Jawaharlal Nehru was first Prime minister of India.
(ii) Sum of two and five is seven.
(iii) Product of a negative and a positive number is a negative number.
(iv) Lucknow is the capital of Uttar Pradesh.
(v) Product of 2 and 3 is 8.
(vi) Area of Madhya Pradesh is less than in the comparion of another states of India.
(vii) Multiplicative Identity of any number is 0.
(viii) Physics is a difficult subject.
(ix) Females are more sensible as compare to Male.
(x) Tomorrow is Thursday.
Here, first four sentences are true. Thus, these are statements. 
(v) to (x) are not true so there are sentences.
Again, consider the following sentence :
‘There are 35 days in the month of July.’
We know that there are 31 days in the month of July. So given sentence is false.
Thus, this is sentence.
Generally, statements are denoted by small letters p, q, r, ....
For example, “Ice is cold” is denoted by p, then p: Ice is cold.

New statements from old Statements
We now look into method for producing new statements from those that we already have. For example, 2 is an even prime number. We can write as “It is false that 2 is not an even prime number”.
Now we will consider an important technique that “what it means to say that a given statement is true or not true.”

Negation of a Statement
The denial of a statement is called the negation of the statement. For example,
(1) Mohan is student of class VII
(2) Mohan is not student of class VII Sentence (2) is negation of sentence (1)
This can simply be expressed as :
“It is false that Mohan is student of class VII” Negation of any simple statement is formed by adding ‘not’ with main verb.
Definition : lip is a statement, then the negation of p is also a statement and is denoted by ~ p, and read as ‘not'.
Let us consider the statement: p : Everyone in India speaks Hindi.
The denial of this statement (Negation) is :
Not everyone in India speaks Hindi.
This does not mean that no person in India speaks Hindi. It says merely that at least one person in India does not speak Hindi.

Compound Statements
A statement that can be formed by combining two or more simple statements is called compound statements and each statement is called a component statement.
For example :
Statement p : ‘36 is divisible by number 4’ is a statement
Statement q : ‘36, is divisible by number 9’ is another statement.
Now combining these two statements a statement can be formed in the following way :
Statement r: ‘36 is divisible by numbers 4 and 9’. Statement r is a compound statement.
Again statement p : ‘ 36 is divisible by number 4 : • True statement 
Then statement r : “36 is divisible by number 4 and 7”—is compound statement.
Here, statements p and q are called component statements and statement r is called compound statement.

The Connective Word 'AND'
Compound statement formed by connective word ‘and’ will be true or false if:
(i) All the component statement of compound statement are true then compound statement will be true.
(ii) All the component statement of compound statement are false or at least one component is false then compound statement is also false.
For example, compound statement
(i) “Diameter of circle is the longest chord and it is twice the length of radius.”
Its component statements are :
P: Diameter of circle is the longest chord. q: Diameter of circle is twice the radius.
Here, both the component statements are true, since both the component statements are true Thus, compound statement is true.

(ii) Number 52 is divisible by numbers 4, 13 and 17. Its component statements are :
p: Number 52 is divisible by number 4 (is true). 
q : Number 52, is divisible by number 13 (is true).
r : Number 52, is divisible by number 17 (is false). 
Here, first two component statements are true but third component statement is false.
Thus, given compound statement is false.
NOTE Do not think that a statement with ‘And" is always a compound statement. For example. A mixture of alcohol and water can be separated by chemical methods. This sentence cannot be considered as a compound statement with "And". Here, the word “And" refers to two things—alcohol and water. Therefore, the word "And" is not used as a connective.

Sentences with the Word 'OR'
Compound statement formed by connective ‘or’ will be true or false, if:
(i) Its one component statement is true or both component statements are true, then compound statement is true. 
(ii) Its both the component statements are false then compound statement is also false.
For example :
p : Number 128, is a multiple of number 32 (true component statement
q : Number 128, is a multiple of number 36 (false component statement)
Then r: Number 128 is multiple of number 32 or 36 (is false) similarly,

(ii) p : Number 33, is multiple of 3 (true component statement)
q: Number 33, is multiple of 11 (True component statement)
Then r: Number 33, is multiple of number 3 or 11 (is true)
Here, in (i) first statement is true and second statement is false. Thus, compound statement is also true, since one component statement is true.
Similarly, In (ii), both the component statements are true. Thus, compound statement is also true, since both the component statement are true.
Again, consider the following statements :
p: Two circles intersect each other at three points
q : Lines joining the intersection points of two circles are tangents of the circles.
r: Two circles intersect each other at three points OR line joining the intersection points of circles are tangents of the circles.
Here, both the component statements p and q are false, Since two circle can not intersect each other at three points along with lines joining the Intersection points cannot be tangents of the circles
Thus, compound statement r is also false, since both the component statements are false.

Exclusive 'OR' and Inclusive 'OR'
Now we will consider the following statement— “Mango shake or butter milk is available with a Thali in a Hotel.”
It means that a person who does not want mango shake can have butter milk along with Thali
or one does not want butter milk can have mango shake along with Thali. But a person cannot have both mango shake and butter-milk.
This is called an exclusive ‘or’
Again, consider the following statement :
“A student who has taken biology or chemistry M.Sc can apply for M.Sc Bio chemistry programme.”
This statement given different meaning as compared to previous one.
 
Here, we mean that the students who have taken both biology and chemistry can apply for the Bio-Chemistry programme, as well as the students who have taken only one of these subjects. In this case we are using Inclusive ‘OR’.
NOTE (i) A compound statement with an ‘Or’ is true when one component statement , is true or both the component1 statements are true.. . ;
(ii) A compound statement with an OR’ is false when both the component statements are false. : '

Quantifiers Phrases
Quantifiers are phrases like, ‘For all’ and ‘There exists’
For example :
(i) There exists a limit whose left hand side and right hand side are equal.
(ii) For all ticket holder journey is legal.
(iii) For each negative number q, √-q is a complex number.

Implications or Conditional Statements
The statement is which ‘if then’ ‘only if’ and ‘if and only if’ are used, called implications or conditional statements. For example ,
Consider the following statement:
“If you are district magistrate, then you are a vip”, we observe that it corresponds to two statements :
p : You are district magistrate.
q : You are a vip. 

Then, the sentence ‘If p then q’:
(i) If p is true, then q must be true.
(ii) If p is false, then it does not say anything on q. 
For example : ‘If you run fast you will reach the train” component of statement are :
p : If you run fast.
q : You can reach the train.
It indicates that p is true then q will necessarily true
i. e. fast running you will reach the train.
If p is false i.e. you are not running fast then it cannot be say that you will not reach the train. (Because train may be late)

Statement : ‘If p, then q’ is same as the following statements :
(1) p implies q is denoted by “p ⇒ q”. The symbol ⇒ stands for implies.
If you are district magistrate, then you are a V.I.P.
(2) p is a sufficient conditon for q.
It means that knowing that you are district magistrate is sufficient to conclude that you are a V.I.P.
(3) p only if q.
It says that you are V.I.P. only if you are district magistrate.
(4) q is a necessary condition for p.
This says that if you are a district magistrate, you are necessarily a V.I.P.
(5) - q implies - p
This says that if your are not a district magistrate, then you are not a V.I.P.

Contrapositive and Converse Statement
Contrapositive statement : If p and q are two statements, then the contrapositive of the implication “If p then q" is “If - q then p”
For example :
1. If a number is divisible by 25, then it is divisible by 5 its implication is as follows :
p : number is divisible by 25.
q : number is divisible by 5.
The contrapositive of this statement is :
If a number is not divisible by 5, it is not divisible by 25.

2. If a triangle is equilateral, then it is also an isoceles its implication is as follows :
p : A triangle is equilateral.
q : A triangle is isoceles.
The contrapositive of this statement is, if a triangle is not isoceles then it is not an equilateral triangle.

(ii) Converse statement : If p and q are two statements, then the converse of the implications “If p then q” is “If? then/?”. -*
For example :
1. If 15 is an odd number, then 3 is its factor. , Converse : If number 3 is a factor of number 15, then 15 is an odd number.
2. “If n is an odd number, then n² will be odd component statements are:
p : n is an odd number. 
q : n² is an odd number.
Converse : “If n² is a odd number, then n is also a odd number.

To Prove Validating Statements
Checking the validity of statement means when it is true and when it is not true. The answer to these questions depends upon which pf the special words and phrases “and”, “or” and which of the implications “If and only if’, “If-then” and which of the quantifiers “for every”, “there exists”, appear in the given statement.
Here, we shall discuss some techniques or rules to find when a statement is valid or true.

Rules for Verification of the Validity of Statements
Rule 1. Statements with ‘And’
If p and q are mathematical statements, then in order to show that the statement ‘p and q' is true, the following steps are followed. ,
Step 1. Show that the statement p is true.
Step 2. Show that the statement q is true.

Rule 2. Statement with ‘or’
If p and q are mathematical statement, then in order to. show that the statement ‘p or q’ is true, one must consider the following.
Case 1: Assuming that p is false, show that q must be true.
Case 2: Assuming that q is false, show that p must be true.

Rule 3. Statement with ‘If-then’
If p and q are two mathematical statement, then 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, prove that q must be' true. (Direct method)
Case 2: By assuming that q is false, prove that p must be false. (Contrapositive)

Rule 4. Statement with ‘If and only if’.'
In order to prove the statement ‘p if and only if q’, we need to show :
(i) If p is true, then q is true.
(ii) If q is true, then p is true.

By Contradiction
Here, to check whether a statement p is true, we assume that p is not true i.e. ~p is true. Then we arrive at some result which contradicts our assumption. Therefore, we conclude that p is true.
By Counter Examples : A method by which we may show that a statement is false. The method involves giving an example of a situation where the statement is not valid. Such an example is called counter example.
NOTE In mathematics, counter examples are used to disprove the statement. However, generating examples in favour of a statement do not provide validity of the statement.

→ Statement: A mathematically acceptable statement is a sentence which is either true or false.

→ Negation of any statement: If p denote a statement, then 'p is not true’is negation of statement p which is denoted by ~ p.

→ Compound statement and their related component statements : 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 role of “AND”, “OR”. “There exists” and “For every” in compound statements.

→ The meaning of implications “If’, “only If’, “If and only if’.
A sentence with if p, then q can be written in the following ways.
(a) p implies q (denoted by p ⇒ q)
(b) p is a sufficient condition for q
(c) q is a necessary condition for p
(d) p only if q
(e) ~ q implies ~ p

→ The contrapositive of a statement p ⇒ q is the statement ~ q ⇒ ~ p

→ The converse of a statement p ⇒ q i& the statement q ⇒ p

→ p ⇒ q together with its converse, gives p if and only if

→ 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.