# Mathematical Reasoning for JEE-Mains

Mathematical Reasoning is a topic covered under the syllabus of JEE-Mains only, excluding JEE-Advanced exam. One question worth 4 marks is asked from this topic in JEE-Mains paper. Generally, students don't pay much attention to this topic especially those who are targeted for JEE-Advanced. So here I'm trying to make it easily covered through this note.

Logic is the subject that deals with the principles of reasoning. Sometimes, we define logic as the science of proof.

$\mathrm{\text{Statements of Logical Sentences}}$

We convey our daily views in the form of sentence which is a collection of words. This group of words is a sentence if it makes some sense.

A declarative sentence, whose truth or falsity can be decided is called a statement of a logical sentence but the sentence should not be imperative, interrogative and exclamatory. A statement is usually denoted by p,q,r or any other small alphabet.

Open Statement:

A sentence which contains one or more variables such that when certain values are given to the variables, it becomes a statement is called an open statement.

Compound Statement:

If two or more simple statements are combined by the use of words such as 'and', 'or', 'not', 'if', 'then', and 'if and only if', then the resulting statement is called a compound statement.

$\mathrm{\text{Truth Value and Truth Table}}$

A statement can either be 'true' or 'false' which are called truth values of a statement and these are represented by the symbols T and F, respectively.

A truth table is a summary of truth values of the resulting statements for all possible assignment of values to the variables appearing in a compound statement.

The number of rows depends on their number of statements.

Truth table for two statements (p,q)

 p q T T T F F T F F

$\mathrm{\text{Logical Operations}}$

The phrases or words which connect simple statements are called logical connectives/operations or sentential connectives or simply connectives.

• AND Operation

A compound sentence formed by two simple sentences p and q using connective 'and' is called the conjunction of p and q. It is represented by $p \wedge q$.

 p q p $\wedge$ q T T T T F F F T F F F F
• OR Operation

A compound statement formed by two simple sentences p and q using connectives 'or' is called disjunction of p and q. It is represented by $p \vee q$.

 p q p $\vee$ q T T T T F T F T T F F F
• Negation/NOT Operation

A statement which is formed by changing the truth value of a given statement by using the word like "no", 'not' is called negation of given statement. If p is a statement, then negation of p is denoted by $\sim p$.

 p $\sim$ p T F T F F T F T
• Conditional Operation

Two simple statements p and q connected by the phrase 'if and then' is called conditional statement of p and q. It is represented by $p \Rightarrow q$.

 p q p $\Rightarrow$ q T T T T F F F T T F F T
• Biconditional Operation

The two simple statements connected by the phrase 'if and only if', this is called biconditional statement. It is denoted by the symbol $\Leftrightarrow$.

 p q p $\Leftrightarrow$ q T T T T F F F T F F F T

$\mathrm{\text{Implications}}$

Students get confused when these four terms are played in their minds: Reverse, Converse, Inverse and Contrapositive.

By definition, the reverse of an implication means the same as the original implication itself. Each implication implies its contrapositive, even intuitionistically. In classical logic an implication is logically equivalent to its contrapositive and moreover, its inverse is logically equivalent to its converse.

Consider the implication formula $p \implies q$.

• Its reverse is $q \Leftarrow p$.

• Its converse is $q \implies p$.

• Its inverse is $\sim p \implies \sim q$.

• Its contrapositive is $\sim q \implies \sim p$.

$\mathrm{\text{Tautology And Contradiction}}$

The compound statement which is true for every value of its components is called tautology. For an example, ( p $\Rightarrow$ q ) $\vee$ ( q $\Rightarrow$ p ) is a tautology.

The compound statement which is false for every value of its components is called contradiction/fallacy. For an example, $\sim$ { ( p $\Rightarrow$ q ) $\vee$ ( q $\Rightarrow$ p )} is a fallacy.

$\text{Truth Table:}$

 p q p $\Rightarrow$ q q $\Rightarrow$ p ( p $\Rightarrow$ q ) $\vee$ ( q $\Rightarrow$ p ) $\sim$ { ( p $\Rightarrow$ q ) $\vee$ ( q $\Rightarrow$ p )} T T T T T F T F F T T F F T T F T F F F T T T F

$\mathrm{\text{Algebra of Statements}}$

• Idempotent Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \vee p \Leftrightarrow p$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \wedge p \Leftrightarrow p$

• Associative Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $(p \vee q ) \vee r \Leftrightarrow p \vee (q \vee r )$

$\qquad \qquad \qquad \qquad \qquad$ 2. $(p \wedge q ) \wedge r \Leftrightarrow p \wedge (q \wedge r )$

• Commutative Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \vee q \Leftrightarrow q \vee p$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \wedge q \Leftrightarrow q \wedge p$

• Distributive Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \vee ( q \wedge r) \Leftrightarrow (p \vee q ) \wedge ( p \vee r)$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \wedge ( q \vee r) \Leftrightarrow (p \wedge q ) \vee ( p \wedge r)$

• Identity Laws

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \wedge T \Leftrightarrow p$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \vee F \Leftrightarrow p$

$\qquad \qquad \qquad \qquad \qquad$ 3. $p \vee T \Leftrightarrow T$

$\qquad \qquad \qquad \qquad \qquad$ 4. $p \wedge F \Leftrightarrow F$

• Complement Laws

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \vee \sim p \Leftrightarrow T$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \wedge\sim p \Leftrightarrow F$

$\qquad \qquad \qquad \qquad \qquad$ 3. $\sim T \Leftrightarrow F$

$\qquad \qquad \qquad \qquad \qquad$ 4. $\sim F \Leftrightarrow T$

• Absorption Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $p \vee ( p \wedge q ) \Leftrightarrow p$

$\qquad \qquad \qquad \qquad \qquad$ 2. $p \wedge ( p \vee q ) \Leftrightarrow p$

• De-Morgan's Law

$\qquad \qquad \qquad \qquad \qquad$ 1. $\sim (p \vee q ) \Leftrightarrow \sim p \wedge \sim q$

$\qquad \qquad \qquad \qquad \qquad$ 2. $\sim (p \wedge q ) \Leftrightarrow \sim p \vee \sim q$

• Involution Law

$\qquad \qquad \qquad \qquad \qquad$ $\sim (\sim p) \Leftrightarrow p$

$\mathrm{\text{Duality}}$

Two compound statements $S_1$ and $S_2$ are said to be duals of each other, if one can be obtained from the other b replacing $\wedge$ by $\vee$ and $\vee$ by $\wedge$. The connectives $\wedge$ and $\vee$ are also called duals of each other.

Symbolically, it can be written as, if $S(p,q)=p \wedge q$, then its dual is $S^*(p,q)=p \vee q$.

$\mathrm{\text{Additional Useful Points}}$

• If a compound statement is made up of n substatements, then its truth value will contain $2^n$ rows.

• A statement which is neither a tautology nor a contradiction is a contingency.

• $p \Rightarrow q = \sim p \vee q$

• $\sim ( p \Rightarrow q) = \sim ( \sim p \vee q) = p \wedge (\sim q)$

• $p \Leftrightarrow q = (p \Rightarrow q) \wedge (q \Rightarrow p)$

• $\sim ( p \Leftrightarrow q ) = (p \wedge \sim q) \vee ( q \wedge \sim p)$

• $( p \Leftrightarrow q) \Leftrightarrow r = p \Leftrightarrow ( q \Leftrightarrow r)$

You can use this note even to discuss the Mathematical Reasoning doubts and problems if you have any. Thanks!

Note by Sandeep Bhardwaj
4 years, 10 months ago

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Great article! I think that it would be useful to many students, especially when we all suffered from the situation with the virus. For example, we now study remotely and this is rather difficult, of course, this https://edubirdie.com/pay-for-research-papers educational platform helps me, but I think I encountered similar difficulties due to the pandemic.

- 8 months ago

Thank you for the best concept

- 2 years, 2 months ago

the best concept indeed lol

- 2 years, 2 months ago

Isn't the laws like what we did in Boolean algebra in the CBSE computer science syllabus?

- 2 years, 9 months ago

Very nice explanation. Could be better... If has examples.

- 2 years, 9 months ago

Thanku so much fr this I went through so many model papers and in every paper there was atleast a question on this topic I used to just skip it cause they didn't do this topic in our college and those symbols ...Seemed complicated... But now...I am able to get my answers Thanku once again

- 2 years, 10 months ago

This is Short and simple. helped me in solving previous years jee mains questions. thanks a lot.

- 2 years, 10 months ago

Thank you for your excellent guide on the mathematical reasoning and it is a very good tool for the preparation of the JEE Mains

- 2 years, 10 months ago

Consider the following statements

P : Suman is brilliant Q : Suman is rich R : Suman is honest

The negation of the statement "Suman is brilliant and dishonest if and only if Suman is rich" can be expressed as

1) (P∧~R)↔QP∧~R↔Q 2) ~(P∧R)↔Q~P∧R↔Q

3) Q↔(P∧~R)Q↔P∧~R

4) ~(Q↔(P∧~R))~Q↔P∧~R A) 1 B) 2 C) 3 D) 4

- 2 years, 10 months ago

4

- 2 years, 1 month ago

- 2 years, 1 month ago

It's okay sandeep but also there are few left in mathematical reasoning that u did not explained please try to cover it .other things are good no problem in it

- 3 years ago

Thank u sandeep.arikatka@gmail.con that was really helpful

- 3 years, 10 months ago

Why and how

- 3 years, 1 month ago

THANK U SIR.Awesome Note sir !

- 3 years, 10 months ago

THANK U SIR.Awesome Note sir !

- 3 years, 10 months ago

could someone please post/guide me to some of the explanations for the laws in the additional useful points? I'm a bit confused in the area related to the implication statements . thanks!

- 4 years ago

Also see my wikis on propositional logic and predicate logic

Additionally, a student interested in Logic should read the book For all x

- 4 years, 9 months ago

no

- 2 years, 2 months ago

THANK U SIR : REVISED MATHEMATICAL REASONING IN JUST 5 MIN ,

- 4 years, 10 months ago

Thanku Sir, your contributions are worth.

- 4 years, 10 months ago

worth....what?

- 2 years, 2 months ago

Can you have some problems of parabola, ellipse,hyperbola for jee mains

- 4 years, 10 months ago

I am trying my best to post maximum problems for JEE. Hopefully I will do within 3-4 days.

- 4 years, 10 months ago

Surely waiting.

- 4 years, 10 months ago

Here's one from my side!

- 4 years, 10 months ago

Nice problem! :)

- 4 years, 10 months ago

Now I'm confident enough to aim for 120!

- 4 years, 10 months ago

Is it you are confident enough to aim for 120 factorial? 😁

- 6 months, 3 weeks ago

@Aditya Kumar In which IIIT you are?

- 2 years, 1 month ago

♥Loved it!

- 4 years, 9 months ago

All the best! I'm confident too that you will get only 120 in Maths only. ;)

- 4 years, 10 months ago

Shut up

- 2 years, 9 months ago

lol

- 2 years, 2 months ago

Awesome Note sir !

- 4 years, 10 months ago

rajdeep bhaiya pls tell me which course shpuld i opt on coursera.org for learning physics for ipho and also share with me if any other website is there

- 3 years, 5 months ago

Thank you :)

- 4 years, 10 months ago

Isn't the truth table of biconditional statements wrong?? Otherwise the note is awesome ... :-)

- 4 years, 10 months ago

Oh yeah (a typo :()! Thanks for pointing it out. Fixed!

- 4 years, 10 months ago

- 4 years, 10 months ago

Thank you! I will spend some time today to improve it further and create a set of problems of this topic for JEE-Mains. :)

- 4 years, 10 months ago

Cool....

- 4 years, 10 months ago