Mathematical Reasoning Chapter 14 Class XI

Complete explanation of chapter 14 class 11 mathematical reasoning. All basic points their explanation with examples

The main topics which are discussed in this chapter are as follows

• Mathematical acceptable statement, difference between a sentence and statement. Negation of a statement, compound statement and their components.
• Special words/phrases, connectives ‘and’ , ‘or’ compound statement with and, compound statement with or, Inclusive ‘or’ exclusive ‘or’
• Quantifiers are the phrases like like : “There exist” , “for all” , “for every” .
• Consolidating the understanding of “If and only if (necessary and sufficient condition” “implies” , “and/or” , “Implied by” , “and” , “or”, “there exist” , and their use through variety of examples related to real life and mathematics.
• Validation of the statement involving the connecting words.
• Explanation of contradiction, converse and contra positive.

Statement: A sentence is called a mathematical acceptable statement if it is either true or false but not both.

For Example:

New Delhi is the capital of India ⇒True ⇒Yes it is a statement.

 New Delhi is the capital of Pakistan ⇒ False  ⇒ Yes it is a statement.

Women are more intelligent than girl ⇒ Sometimes it is true and sometimes it is false  ⇒  It is not a statement. It is a simple sentence.

Following Sentences are never be the statement

(i) Exclamatory (!) sentences

(ii) order-able sentences

(iii) interrogative (?)Sentences

(iv) Sentences involving time: today, tomorrow, and yesterday

(v) Sentences which involves the terms  he, she, it, you, here, there etc.

(vi) Sentences which involves the terms here, there, everywhere etc.

Negation (or Denial) of a statement

The denial of a statement is called negation of the statement.

Consider the sentence  p: New Delhi is a city.

Negation of this statement is :

 It is not the case that New Delhi is a city  or

It is false that New Delhi is a city or

New Delhi is not a city.

Note: If p is a statement then negation of p is also a statement and is denoted by ∽ p and read as  'not p'

While writing the negation of the given statement we use the words 

 “It is not the case” or “ It is false that” or simply using not with the helping verb .

Compound Statement

Compound statements are of two types (1) Conjunction (2) Disjunction

(1) Conjunction : If two statements are combined by the connective word 'and' then the compound statement so formed is called the 'conjunction of the original statement.

For Example:  p: Ravi is a boy,   q ; Ambika is a girl

Conjunction of p and q is given by

p ∧ q = Ravi is a boy and Ambika is a girl

A compound statement with “and” is true if all its component statements are true.

or p ∧ q is true when both p and q are true

A compound statement with ‘and’ is false if any of its compound statement is false.

Note : Do not think that a statement with ‘and’ is always a compound statement. Example : A mixture of alcohol and water can be separated by chemical method. In this statement and is not act as connective and it is only a one statement.

(2) Disjunction : If two statements are combined by the connective word 'or' then the compound statement so formed is called the 'disjunction' of the original statement.

For Example: p: There is something wrong with the teacher.  q : There is something wrong with the student.

Disjunction of p and q is given by

p ⅴ q = There is something wrong with the teacher or with the student.

A compound statement with ‘or’ is true when one component statement is true or both the component statements are true.

A compound statement with ‘or’ is false only when both the component statements are false.

or  p ⅴ q is false when both p and q are false.

Inclusive ‘or’ :

Example : A student who has taken Biology or Chemistry can apply for M.Sc. microbiology. In this statement ‘or’ is inclusive.

Because if a student have Biology can apply for M. Sc. microbiology.

If a student have Chemistry can apply for M.Sc. microbiology.

Exclusive ‘or’ : 

Example : An ice-cream or pepsi is available with a thali in a restaurant. Here ‘or’ is exclusive. In this statement ‘or’ is inclusive.

A person can either take  ice-cream or pepsi with a thali but cannot take both. So here 'or' is exclusive.

Quantifiers : Quantifiers are the phrase like : “There exist” , “for all” , “for every” 

Use of these words in different examples.

For Example:

There exists a rectangle whose all sides are equal.

For all parallelograms opposite sides are equal and parallel.

Implications : There are many statements which contains the word  like : ‘If-then’ ,    ‘only if ’ , ‘if and only if’ ,  such statements are called Implications.

Example: If you get a job then your credentials are good.

Contra-positive of a statement : It is the method of righting the reverse of a given statement with  negation.

If p and q are two statement then the contra positive of the implication : "if p the q" is "if ∽q, then ∽p"

Example: If a number is divisible by 9 then it is divisible by 3

Contra positive:  If a number is not divisible by 3 then it is not divisible by 9.

Converse of a statement: It is the method of writing the reverse of the statement.

Example: If a number is divisible by 9 then it is divisible by 3

Converse : If a number is divisible by 3 then it is divisible by 9.

Inverse of a Statement: If p and q are two statements, then inverse of "if p then q" is "if ∽p, then ∽q"

Example: If a number is divisible by 9 then it is divisible by 3

Converse : If a number is not divisible by 9, then it is not divisible by 3.

Methods of proving and disproving the given statement

There are three methods

Method of contradiction : It is the method of proving the given statement by taking counter example.

Method of contra positive

Method of converse