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)

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

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 $p \wedge q$.

• 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 $p \vee q$.

• 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 $\sim p$.

• 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 $p \Rightarrow q$.

• 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 $\Leftrightarrow$.

$\mathrm{\text{Implications}}$

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.

$\text{Truth Table:}$

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

• 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!