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### Mathematical Reasoning Chapter 14 Class XI

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# Mathematical Reasoning Chapter 14 Class XI

**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.**

**Sta**

**tement: 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 stat****ement**

**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**

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