JEE 2022 | Class 12

JEE - Mathematical Reasoning

Get interactive courses taught by top teachers
Introduction

Introduction

Mathematical reasoning is the concept in Mathematics that deals with finding the truth values of any Mathematical statement.  The principle of Mathematical reasoning is generally used to analyse the conceptual logical thinking capacity of an individual in competitive examinations and eligibility tests. Mathematical reasoning questions are extremely interesting and successfully stir up the rational thinking of the human brain. There are various kinds of statements in Mathematical reasoning and the operations that are performed on those statements.


Important Terminologies in Mathematical Reasoning

  • Statement

  • Truth value of a statement 

  • Truth Table

  • Negation operation

  • Compound statements

  • Conditional Statements

  • Contrapositive Statement

  • Tautology

  • Algebra of Statements


Compound Statement and Basic logical Statement

If two or more simple statements are combined by the words ‘and’, ‘or’, ‘if…..then’, ‘if and only if’ then the resultant statement is called a compound statement and words ‘and’, ‘or’ , ‘if…..then’, ‘if and only if’ are called logical connectives.


Examples: (i) Yuvi studies in class XII and resides in Himachal Pradesh.


(ii) Vineet might be watching a movie or playing chess

(iii) India can win the match if and only if Sachin hits a six on each of the remaining 3 balls.


Basic Logical Connective 

Symbol 

Compound Statement

Example

And

$\wedge$

Conjunction

$p\wedge q$

Or

$\vee$

Disjunction

$p\vee  q$

If…..then

$\rightarrow$

Conditional statement

$p\rightarrow q$

If and only if

$\leftrightarrow$

Biconditional statement

$p\leftrightarrow  q$ 


Truth Table of Compound Statements Using Connective ‘AND’ and ‘OR’

Truth Values of statement ‘ p and q’ and ‘p or q’ are tabulated as follows:


$p$

$q$

$p\wedge q$

$p\vee  q$

T

T

T

T

T

F

F

T

F

T

F

T

F

F

F

F


  • Statement ‘p and q’ is true when both p and q are true. Also, this statement is false if at least one of the statements is false.

  • Statement ‘p or q’ is true if at least one of the statements is true. Also, this statement is false if both statements are false.


Conditional Statements

Consider the following compound statements:

If Varanasi is in India then 3 + 6 = 9.

If Varanasi is in India then 3 + 6 = 11

If Varanasi is in China then 3 + 6 = 9.

If Varanasi is in India then 3 + 6 = 11.

If the first statement is p and the second statement is q, then we have the following truth table for $p\rightarrow q$.


$p$

$q$

$p\rightarrow q$

T

T

T

T

F

F

F

T

T

F

F

T


Biconditional Statements:

Consider the following compound statements

If Varanasi is in India if and only if 3 + 6 = 9.

If Varanasi is in India if and only if 3 + 6 = 11

If Varanasi is in China if and only if 3 + 6 = 9.

If Varanasi is in India if and only if 3 + 6 = 11.


If the first statement is p and the second statement is q, then the following truth table for $p\leftrightarrow  q$.


$p$

$q$

$p\leftrightarrow  q$

T

T

T

T

F

F

F

T

F

F

F

T


Logical Equivalence

Two compound statements $S_{1}\left ( p,\,q,\,r,\,.... \right )$ and $S_{2}\left ( p,\,q,\,r,\,.... \right )$ of component statement p, q, r …. are called logically equivalent or simply equivalent or equal if they have identical truth values and we can write 

$\Rightarrow S_{1}\left ( p,\,q,\,r,\,.... \right )\equiv S_{2}\left ( p,\,q,\,r,\,.... \right )$

Note:  

  • $p\to q\equiv \left ( \sim p \right )\vee q$

  • $p\leftrightarrow q\equiv \left ( p\to q \right )\wedge \left ( q\rightarrow p \right )$

  • $p\leftrightarrow q\equiv \left ( \sim p\vee q \right )\wedge \left ( p\,\,\vee \sim q \right )$


Converse, Inverse and Contrapositive of Statement

Given an if-then statement “if p, then q” we can create three related statements.

A conditional statement consists of two parts, a hypothesis in the “ if'' clause and a conclusion in the “then” clause.

For instance, consider the compound statement, 

“If it rains, then they cancel the picnic.”

Here, 

“It rains” is the hypothesis.

“They cancel the picnic” is the conclusion.

  • To form the converse of the conditional statement, just interchange the hypothesis and the conclusion. The converse of the statement (1) is “if they cancel the picnic, it rains.”

  • To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of the statement (1) is “if it does not rain, then they do not cancel the picnic.”

  • To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of the statement (1) is “If they do not cancel the school, then it does not rain.”


Statement 

If p then q $\left ( p\rightarrow q \right )$

Converse

If q then p $\left ( q\rightarrow p \right )$

Inverse

If not p then not q $\left ( \sim p\rightarrow \sim q \right )$

Contrapositive

If not q then not p $\left ( \sim q\rightarrow \sim p \right )$


Tautology and Contradiction (Fallacy)

  • Tautology: A compound statement is called a tautology if it is always true for all possible truth values of its component statements

  • Contradiction: A compound statement is called a contradiction if it is always false for all possible values of its component statement. A contradiction is also called a fallacy.

  • The negation of a tautology is a contradiction and negation of contradiction is a tautology.


Algebra of Statements

  • Idempotent Laws: If p is any statement, then 

(i). $p\vee p\equiv p$            (ii) $p\wedge  p\equiv p$

  • Complement Laws: If p is any statement, then

(i). $p\,\vee \sim p\equiv t$       (ii) $p\,\,\wedge \sim p\equiv f$    

(iii) $\sim\sim p\equiv p$           (iv) $\sim t\equiv f$ 

(v) $\sim f\equiv t$ 

  • Identity Laws: If p is any statement, then 

(i) $p\wedge  t\equiv p$              (ii) $p\wedge  f\equiv f$

(iii) $p\vee t\equiv t$                   (iv) $p\vee f\equiv p$

  • Commutative Laws: If p and q be any two statements then

(i) $p\wedge q\equiv q\wedge p$     (ii) $p\vee q\equiv q\vee  p$

  • De Morgan’s Laws: If p and q be any two statements, then 

(i) $\sim\left ( p\vee q \right )\equiv \sim p\,\,\wedge \sim q$ 

(ii) $\sim\left ( p\,\ \wedge  q \right )\equiv \sim p\,\,\vee  \sim q$ 

  • Associative Laws: If p, q and r are any three statements, then 

(i) $\left ( p\wedge q \right ) \wedge r\equiv p\wedge \left ( q\wedge r \right )$

(ii) $\left ( p\vee q \right )\vee r\equiv p\vee \left ( q\vee r \right )$

  • Distributive Laws: If p, q and r are any three statements, then

(i) $p\wedge \left ( q\vee r \right )\equiv \left ( p\wedge q \right )\vee \left ( p\wedge r \right )$

(ii) $p\vee \left ( q\wedge r \right )\equiv \left ( p\vee q \right )\wedge \left ( p\vee r \right )$


Solved Examples:

Example 1: Show that the compound statements $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ are logically equivalent.

Solution: Truth values of $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ are as follows:


$p$

$q$

$\sim p$

$p\vee q$

$\left (  p\vee q\right )\wedge \sim p$

$\sim p\,\,\wedge q$

T

T

F

T

F

F

T

F

F

T

F

F

F

T

T

T

T

T

F

F

T

F

F

F


Clearly, $\left ( p\vee q \right )\wedge \sim p$ and $\sim p \wedge q$ have identical truth values.

$\therefore \left ( p \vee q \right )\wedge \sim p \equiv \,\,\sim p\wedge q$ 


Example 2: Show that $p\to \left ( p\vee q \right )$ is a tautology

Solution: Truth values of $p\to \left ( p\vee q \right )$

$p$

$q$

$p\vee q$

$p\to \left ( p\vee q \right )$

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

T


Thus, for all possible truth values of p and q, the compound statement $p\to \left ( p\vee q \right )$ is true.


Example 3: Prove that $q\,\,\wedge \sim p\equiv \sim \left ( q\to p \right )$

Solution: $\sim\left ( q\to p \right ) \equiv \,\sim \left ( \sim q\vee p \right )$

$\Rightarrow \sim\left ( q\to p \right ) \equiv \,\sim \left ( \sim q \right )\wedge \sim p$

$\Rightarrow \sim\left ( q\to p \right ) \equiv \, q\,\,\wedge \sim p$


Solved problems of Previous Year Question

1. Which of the following is tautology?

  1. $A\wedge \left ( A\to B \right )\to B$

  2. $B\to \left ( A\wedge A\to B \right )$

  3. $A\wedge \left ( A\vee B \right )$

  4. $\left (A\vee B  \right )\wedge A$

Ans: (a)

$\Rightarrow A\wedge \left ( A\to B \right ) \to B$

$= A\wedge \left ( \sim A\vee B \right )\to B$

$=\left [ \left ( A\wedge \sim A \right )\vee \left ( A\wedge B \right ) \right ]\to B$

$=\left ( A\wedge B \right )\to B$

$=\,\, \sim A\,\,\vee \sim B\vee B$

$= t$


2. The Boolean expression $\sim\left ( p\vee q \right )\vee \left ( \sim p\vee q \right )$ is equivalent to 

  1. $\sim q$

  2. $\sim p$

  3. $p$

  4. $q$

Ans: (b)

$\Rightarrow \,\,\sim \left ( p\vee q \right )\vee \left ( \sim p\wedge q \right )$

$= \left ( \sim p\,\,\wedge \sim q \right )\vee \left ( \sim p\wedge q \right )$

$=\sim p\wedge \left ( \sim q\wedge q \right )$

$=\,\,\sim p$


3. The statement $\sim \left ( p\leftrightarrow \,\sim q \right )$ is 

  1. equivalent to $p\leftrightarrow q$

  2. equivalent to $\sim p\leftrightarrow q$

  3. a tautology 

  4. a fallacy

Ans: (a)


$p$

$q$

$\sim q$

$p\leftrightarrow \,\,\sim q$

$\sim\left ( p\leftrightarrow \,\,\sim q \right )$

$p\leftrightarrow q$

T

T

F

F

T

T

T

F

T

T

F

F

F

T

F

T

F

F

F

F

T

F

T

T

 

Practice Question:

1. $\sim\left ( p\vee \left ( \sim p\vee q \right ) \right )$ is equal to

  1. $\sim p\wedge \left ( p\,\,\wedge \sim q \right )$

  2. $\left ( p\,\,\wedge \sim q \right )\vee \sim p$

  3. $\left ( p\,\,\vee \sim q \right )\vee \sim p$

  4. None of these


2. $\left ( p\,\,\wedge \sim q \right )\wedge \left ( \sim p\wedge q \right )$ is 

  1. A tautology

  2. A contradiction

  3. Neither a tautology nor a contradiction

  4. None of these


3. If $p\to \left ( q\vee r \right )$ is false, then the truth values of p, q and r are, respectively

  1. T, T, F

  2. F, F, F

  3. F, T, T

  4. T, F, F


Answer: 

  1. (a)

  2. (b)

  3. (d) 


Conclusion:

In this chapter, we have elaborated on concepts and solutions to questions on the topic of Mathematical reasoning.  Everything you're looking for is available in a single location. Students can carefully read through the Concepts, Definitions, and questions in the PDFs, which are also free to download and understand the concepts used to solve these questions. This will be extremely beneficial to the students in their exams.

See More
JEE Main Important Dates

JEE Main Important Dates

View all JEE Main Exam Dates
JEE Main 2022 June and July Session exam dates and revised schedule have been announced by the NTA. JEE Main 2022 June and July Session will now be conducted on 20-June-2022, and the exam registration closes on 5-Apr-2022. You can check the complete schedule on our site. Furthermore, you can check JEE Main 2022 dates for application, admit card, exam, answer key, result, counselling, etc along with other relevant information.
See More
View all JEE Main Exam Dates
JEE Main Information

JEE Main Information

Application Form
Eligibility Criteria
Reservation Policy
Admit Card
NTA has announced the JEE Main 2022 June session application form release date on the official website https://jeemain.nta.nic.in/. JEE Main 2022 June and July session Application Form is available on the official website for online registration. Besides JEE Main 2022 June and July session application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details. July Session's details will be updated soon by NTA.
JEE Main 2022 applicants should be aware of the eligibility criteria before applying to the exam. NTA has released all the relevant information on the official website, i.e. https://jeemain.nta.nic.in/. JEE Main 2022 aspirants should have passed Class 12th or any other equivalent qualifying examination in 2021, 2020, or students appearing in the Class 12th final exam in 2022 can also apply. For further details, visit our website.
As per the union government’s norms, NTA has released the JEE Main 2022 June and July session reservation criteria for different candidates’ categories (SC/ST/OBC/PwD), All India Quota, State Government Quota, Deemed Universities, and more. You can check more details on NTA website.
NTA will release the admit card for JEE Main 2022 June and July Session at https://jeemain.nta.nic.in/, 15 days prior to the exam date for the registered candidates. The admit card will contain information such as the name and contact details of the candidate, the exam centre, reporting time, and examination schedule along with other important instructions for JEE Main 2022 June and July Session.
It is crucial for the the engineering aspirants to know and download the JEE Main 2022 syllabus PDF for Maths, Physics and Chemistry. Check JEE Main 2022 syllabus here along with the best books and strategies to prepare for the entrance exam. Download the JEE Main 2022 syllabus consolidated as per the latest NTA guidelines from Vedantu for free.
See More
Download full syllabus
Download full syllabus
View JEE Main Syllabus in Detail
JEE Main 2022 Study Material

JEE Main 2022 Study Material

View all study material for JEE Main
JEE Main 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Main Study Materials. It covers the entire JEE Main syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Main study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Main exam 2022 with Vedantu right on time.
See More
All
Mathematics
Physics
Chemistry
See All
JEE Main Question Papers

JEE Main Question Papers

see all
Download JEE Main Question Papers & ​Answer Keys of 2021, 2020, 2019, 2018 and 2017 PDFs. JEE Main Question Paper are provided language-wise along with their answer keys. We also offer JEE Main Sample Question Papers with Answer Keys for Physics, Chemistry and Maths solved by our expert teachers on Vedantu. Downloading the JEE Main Sample Question Papers with solutions will help the engineering aspirants to score high marks in the JEE Main examinations.
See More
JEE Main 2022 Book Solutions and PDF Download

JEE Main 2022 Book Solutions and PDF Download

View all JEE Main Important Books
In order to prepare for JEE Main 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, RS Aggarwal Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Main 2022 exam so that they can grab the top rank in the all India entrance exam.
See More
Maths
NCERT Book for Class 12 Maths
Physics
NCERT Book for Class 12 Physics
Chemistry
NCERT Book for Class 12 Chemistry
Physics
H. C. Verma Solutions
Maths
R. D. Sharma Solutions
Maths
R.S. Aggarwal Solutions
See All
JEE Main Mock Tests

JEE Main Mock Tests

View all mock tests
JEE Main 2022 free online mock test series for exam preparation are available on the Vedantu website for free download. Practising these mock test papers of Physics, Chemistry and Maths prepared by expert teachers at Vedantu will help you to boost your confidence to face the JEE Main 2022 examination without any worries. The JEE Main test series for Physics, Chemistry and Maths that is based on the latest syllabus of JEE Main and also the Previous Year Question Papers.
See More
JEE Main 2022 Cut-Off

JEE Main 2022 Cut-Off

JEE Main Cut Off
NTA is responsible for the release of the JEE Main 2022 June and July Session cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Main 2022 June and July Session is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Main qualifying marks for 2021 ranged from 87.8992241 for general-category, while for OBC/SC/ST categories, they ranged from 68.0234447 for OBC, 46.8825338 for SC and 34.6728999 for ST category.
See More
JEE Main 2022 Results

JEE Main 2022 Results

JEE Main 2022 June and July Session Result - NTA has announced JEE Main result on their website. To download the Scorecard for JEE Main 2022 June and July Session, visit the official website of JEE Main NTA.
See More
Rank List
Counselling
Cutoff
JEE Main 2022 state rank lists will be released by the state counselling committees for admissions to the 85% state quota and to all seats in NITs and CFTIs colleges. JEE Main 2022 state rank lists are based on the marks obtained in entrance exams. Candidates can check the JEE Main 2022 state rank list on the official website or on our site.
The NTA will conduct JEE Main 2022 counselling at https://josaa.nic.in/. There will be two rounds of counselling for admission under All India Quota (AIQ), deemed and central universities, NITs and CFTIs. A mop-up round of JEE Main counselling will be conducted excluding 15% AIQ seats, while the dates of JEE Main 2022 June and July session counselling for 85% state quota seats will be announced by the respective state authorities.
NTA is responsible for the release of the JEE Main 2022 June and July Session cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Main 2022 June and July Session is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Main qualifying marks for 2021 ranged from 87.8992241 for general category, while for OBC/SC/ST categories, they ranged from 68.0234447 for OBC, 46.8825338 for SC and 34.6728999 for ST category.
Want to know which Engineering colleges in India accept the JEE Main 2022 scores for admission to Engineering? Find the list of Engineering colleges accepting JEE Main scores in India, compiled by Vedantu. There are 1622 Colleges that are accepting JEE Main. Also find more details on Fees, Ranking, Admission, and Placement.
See More
question-image

FAQs on JEE - Mathematical Reasoning

FAQ

1. What is a fallacy in mathematical reasoning?

Fallacies are errors in assumptions that are induced by logical inaccuracy.

2. What is a boolean expression?

Boolean Algebra is the branch of mathematics that works with only binary values. In contrast to numerical operations like addition and subtraction, a Boolean equation deals with conjunction, disjunction, and negation. Boolean algebra uses binary codes to express the value and carries out logical computations through operations like AND and OR. The simplicity of Boolean expression makes it the best choice for being used in electronics and digital advancements. It would not be wrong to say that it has successfully revolutionised the world of computers.

3. What are Mathematical Statements?

A mathematical statement is the basis of all mathematics quantitative reasoning. Further, reasoning can be either inductive (also called mathematical induction) or deductive. An assertive statement that can either be true or false is said to be an acceptable statement in mathematics. It cannot be both. Ambiguous statements are invalid in maths and reasoning.