What is the difference between converse and contrapositive?

For implication p → q: Converse is q → p (swap hypothesis and conclusion). Contrapositive is ~q → ~p (swap and negate both). The contrapositive is logically equivalent to the original implication (p → q ≡ ~q → ~p), but the converse is NOT equivalent to the original.

Mathematical Reasoning JEE Notes 2026 | Statements, Tautology, Contrapositive PYQs

1

Statements & Propositions

The fundamental building block of mathematical reasoning is the concept of a statement (also called a proposition). Without statements, we cannot construct any logical argument. Understanding what qualifies as a statement is the first and most important step in this chapter.

1.1 Definition of a Statement

Most Repeated PYQ — Definition of Statement

A statement (or proposition) is a declarative sentence that is either true or false, but not both simultaneously.

Key characteristics:

It must be a declarative sentence (not a question, command, or exclamation).
It must have a definite truth value — either True (T) or False (F).
The truth value must be uniquely determined (not both T and F at the same time).
It need not be known whether the statement is true or false — only that it has a definite truth value.

1.2 Examples of Statements

Valid statements:

"2 + 2 = 4" → Statement (True)
"5 is a prime number." → Statement (True)
"The sun rises in the west." → Statement (False)
"Every even number greater than 2 is the sum of two primes." (Goldbach's conjecture) → Statement (truth value definite but unknown)
"π is an irrational number." → Statement (True)
"3 > 5" → Statement (False)

NOT statements:

"What time is it?" → Interrogative (no truth value)
"Close the door." → Imperative (command, no truth value)
"How beautiful!" → Exclamatory (no truth value)
"x + 1 = 5" → Open sentence (depends on value of x)
"He is a good person." → Subjective (depends on interpretation)
"Today is Monday." → Not a statement in isolation (depends on which day)

1.3 Simple and Compound Statements

Most Repeated PYQ — Types of Statements

Simple Statement (Atomic Statement): A statement that cannot be broken down into simpler statements.

Examples: "3 is odd.", "π > 3."

Compound Statement: A statement formed by combining two or more simple statements using logical connectives (like AND, OR, NOT, IF-THEN).

Examples: "3 is odd AND π > 3.", "If it rains, then I will stay home."

1.4 Components of a Compound Statement

A compound statement consists of:

Component statements: The simpler statements that are combined.
Logical connectives: Words/symbols like AND (∧), OR (∨), NOT (~), IF-THEN (→), IF AND ONLY IF (↔).

Example: In "It is raining and it is cold", the component statements are "It is raining" and "It is cold", and the connective is "AND".

PYQ — JEE Main 2019 (Jan 12 Shift 1)

Question: Which of the following is not a statement?

(a) 8 is greater than 6

(b) Every set is a finite set

(c) The sum of two odd numbers is even

(d) Listen to me carefully

Solution:

A statement must be declarative with definite truth value.

(a) "8 is greater than 6" → True → Statement ✓

(b) "Every set is a finite set" → False (infinite sets exist) → Statement ✓

(c) "Sum of two odd numbers is even" → True → Statement ✓

(d) "Listen to me carefully" → Imperative sentence, no truth value → NOT a statement ✗

\text{Answer: (d) Listen to me carefully}

Answer: (d) Listen to me carefully

2

Truth Values & Truth Tables

Every statement has a truth value — either True (T) or False (F). A truth table systematically lists all possible combinations of truth values of component statements and the resulting truth value of the compound statement. Truth tables are essential tools in mathematical reasoning.

2.1 Truth Values

Most Repeated PYQ — Truth Values

The truth value of a statement is:

T (True) if the statement is correct/factual
F (False) if the statement is incorrect

Truth values are often denoted by: T/F, 1/0, or ✓/✗.

2.2 Construction of Truth Tables

Most Repeated PYQ — Number of Rows in Truth Table

If a compound statement involves n simple component statements, the truth table has:

2^n \text{ rows}

Examples:

1 component (p): 2¹ = 2 rows
2 components (p, q): 2² = 4 rows
3 components (p, q, r): 2³ = 8 rows
4 components (p, q, r, s): 2⁴ = 16 rows

2.3 Standard Pattern for Truth Table with Two Variables

Row	p	q
1	T	T
2	T	F
3	F	T
4	F	F

Memory Trick for Two Variables

For the first variable (p): write T twice, then F twice (T, T, F, F).

For the second variable (q): alternate T, F, T, F.

For three variables (p, q, r): p has 4 T's then 4 F's; q has 2 T's, 2 F's, 2 T's, 2 F's; r alternates T, F, T, F, T, F, T, F.

PYQ — JEE Main 2014 (Apr 19)

Question: The number of rows in the truth table of a compound statement with 4 component statements is:

(a) 8

(b) 12

(c) 16

(d) 32

Solution:

Number of rows = 2ⁿ where n = number of component statements.

Here n = 4, so number of rows = 2⁴ = 16.

2^4 = 16 \text{ rows}

Answer: (c) 16

3

Logical Connectives (NOT, AND, OR)

Logical connectives (also called logical operators) are symbols or words used to combine simple statements into compound statements. The five basic connectives are: NOT (negation), AND (conjunction), OR (disjunction), IF-THEN (implication/conditional), and IF AND ONLY IF (biconditional).

3.1 Negation (NOT)

Most Repeated PYQ — Negation

The negation of a statement p, denoted by ~p (or ¬p, p', or "NOT p"), is the statement that is true when p is false and false when p is true.

\text{If } p \text{ is T, then } \sim p \text{ is F.}\] \[\text{If } p \text{ is F, then } \sim p \text{ is T.}

Truth table:

p	~p
T	F
F	T

Example: p: "2 is greater than 3" (F). ~p: "2 is not greater than 3" (T).

Double negation: ~(~p) ≡ p (Involution Law).

3.2 Conjunction (AND)

Most Repeated PYQ — Conjunction

The conjunction of two statements p and q, denoted by p ∧ q (read as "p AND q"), is the compound statement that is true only when both p and q are true.

p \wedge q \text{ is T} \iff p \text{ is T AND } q \text{ is T}

Truth table:

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Example: p: "3 is odd", q: "5 is prime". p ∧ q: "3 is odd and 5 is prime" (T ∧ T = T).

3.3 Disjunction (OR)

Most Repeated PYQ — Disjunction

The disjunction of two statements p and q, denoted by p ∨ q (read as "p OR q"), is the compound statement that is false only when both p and q are false; it is true if at least one of p, q is true.

p \vee q \text{ is F} \iff p \text{ is F AND } q \text{ is F}

Note: This is inclusive OR (allows both to be true), not exclusive OR (XOR).

Truth table:

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

Example: p: "4 is even", q: "4 is prime". p ∨ q: "4 is even or 4 is prime" (T ∨ F = T).

3.4 Exclusive OR (XOR)

The exclusive disjunction (XOR), denoted p ⊕ q or p ⊻ q, is true when exactly one of p, q is true (but not both).

p \oplus q \equiv (p \vee q) \wedge \sim(p \wedge q)

Example: "Either today is Monday or today is Tuesday" (cannot be both).

3.5 Summary Table of Connectives

Connective	Symbol	Name	Read as	True when
NOT	~ (or ¬)	Negation	"not p"	p is false
AND	∧	Conjunction	"p and q"	both p, q true
OR	∨	Disjunction	"p or q"	at least one true
IF-THEN	→	Implication	"if p then q"	(see Ch. 4)
IFF	↔	Biconditional	"p iff q"	both same value

PYQ — JEE Main 2020 (Sep 2 Shift 1)

Question: If p: "It is raining" and q: "It is cloudy", then the statement "It is raining and it is cloudy" is represented as:

(a) p ∨ q

(b) p ∧ q

(c) p → q

(d) ~p ∧ q

Solution:

"It is raining AND it is cloudy" → uses the AND connective.

AND is denoted by ∧ (conjunction).

p \wedge q

Answer: (b) p ∧ q

4

Conditional & Biconditional Statements

The conditional (implication, IF-THEN) and biconditional (IF AND ONLY IF) are two of the most important connectives in mathematical reasoning. They form the basis of mathematical theorems and proofs.

4.1 Conditional (Implication, IF-THEN)

Most Repeated PYQ — Implication

The conditional (or implication) of p and q, denoted by p → q (read as "if p then q" or "p implies q"), is the compound statement that is false only when p is true and q is false; otherwise it is true.

p \to q \text{ is F} \iff p \text{ is T AND } q \text{ is F}

Truth table:

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Key insight: If the hypothesis (p) is false, the implication is vacuously true ("ex falso quodlibet" — from false, anything follows).

Terminology: In p → q, p is called the antecedent (hypothesis) and q is called the consequent (conclusion).

4.2 Important Equivalent Form of Implication

Most Repeated PYQ — p → q ≡ ~p ∨ q

The implication p → q is logically equivalent to the disjunction:

p \to q \equiv \sim p \vee q

This is a very important equivalence used frequently in JEE.

Example: "If it rains, then I will carry an umbrella" ≡ "Either it does not rain, or I will carry an umbrella."

4.3 Biconditional (IF AND ONLY IF)

Most Repeated PYQ — Biconditional

The biconditional of p and q, denoted by p ↔ q (read as "p if and only if q" or "p iff q"), is true when both p and q have the same truth value (both T or both F).

p \leftrightarrow q \text{ is T} \iff p \equiv q \text{ (same truth value)}

Truth table:

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

Important equivalence:

p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)

4.4 Order of Precedence of Connectives

The order of precedence (from highest to lowest) is:

~ (Negation) — highest
∧ (Conjunction)
∨ (Disjunction)
→ (Implication)
↔ (Biconditional) — lowest

Example: ~p ∧ q → r is interpreted as ((~p) ∧ q) → r.

PYQ — JEE Main 2015 (Apr 10)

Question: The statement p → q is logically equivalent to:

(a) p ∨ q

(b) ~p ∨ q

(c) p ∨ ~q

(d) ~p ∧ q

Solution:

The implication p → q is equivalent to ~p ∨ q.

Verification by truth table:

p	q	~p	p→q	~p∨q
T	T	F	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

Columns p→q and ~p∨q match → equivalent.

p \to q \equiv \sim p \vee q

Answer: (b) ~p ∨ q

5

Converse, Inverse & Contrapositive

Given an implication p → q, we can form three related implications: the converse, inverse, and contrapositive. These are extremely important for JEE and appear in nearly every exam session on Mathematical Reasoning.

5.1 Definitions

Most Repeated PYQ — Converse, Inverse, Contrapositive

For the implication p → q:

Name	Statement	Equivalence
Converse	q → p	≡ Inverse
Inverse	~p → ~q	≡ Converse
Contrapositive	~q → ~p	≡ Original (p → q)

Key results:

p → q ≡ ~q → ~p (Contrapositive law) — most important!
Converse ≡ Inverse (q → p ≡ ~p → ~q)
Neither converse nor inverse ≡ original

5.2 Verification by Truth Table

p	q	~p	~q	p→q	q→p	~p→~q	~q→~p
T	T	F	F	T	T	T	T
T	F	F	T	F	T	T	F
F	T	T	F	T	F	F	T
F	F	T	T	T	T	T	T

Observe the Columns

p→q column matches ~q→~p column → Contrapositive ≡ Original ✓
q→p column matches ~p→~q column → Converse ≡ Inverse ✓
p→q column ≠ q→p column → Original ≠ Converse ✗

5.3 Worked Examples

Example: p: "It is a square", q: "It is a rectangle".

Original (p → q): "If it is a square, then it is a rectangle." (True)

Converse (q → p): "If it is a rectangle, then it is a square." (False)

Inverse (~p → ~q): "If it is not a square, then it is not a rectangle." (False)

Contrapositive (~q → ~p): "If it is not a rectangle, then it is not a square." (True)

PYQ — JEE Main 2023 (Apr 13 Shift 1)

Question: The contrapositive of the statement "If I am not feeling well, then I will go to the doctor" is:

(a) If I am feeling well, then I will not go to the doctor

(b) If I will go to the doctor, then I am not feeling well

(c) If I will not go to the doctor, then I am feeling well

(d) If I am feeling well, then I will go to the doctor

Solution:

Let p: "I am not feeling well", q: "I will go to the doctor".

Original: p → q.

Contrapositive: ~q → ~p.

~q: "I will not go to the doctor".

~p: "I am feeling well" (note: "not (not feeling well)" = "feeling well").

So contrapositive: "If I will not go to the doctor, then I am feeling well."

\text{If I will not go to the doctor, then I am feeling well.}

Answer: (c) If I will not go to the doctor, then I am feeling well

6

Tautology, Contradiction & Contingency

Based on the truth values of a compound statement across all possible combinations, we classify it as a tautology, contradiction, or contingency. This classification is crucial for identifying logical truths.

6.1 Tautology

Most Repeated PYQ — Tautology

A compound statement is a tautology if it is always true, regardless of the truth values of its component statements.

\text{Tautology: all T's in final column of truth table}

Examples of tautologies:

p ∨ ~p (Law of Excluded Middle)
p → p
(p → q) ∨ (q → r)
p → (q → p)
~(p ∧ ~p) (Law of Non-Contradiction)

6.2 Contradiction

Most Repeated PYQ — Contradiction

A compound statement is a contradiction (or absurdity) if it is always false, regardless of the truth values of its components.

\text{Contradiction: all F's in final column of truth table}

Examples of contradictions:

p ∧ ~p (Law of Contradiction)
~(p ∨ ~p)
(p → q) ∧ (p ∧ ~q)

6.3 Contingency

A compound statement is a contingency if it is neither a tautology nor a contradiction — i.e., it is true for some combinations and false for others.

Examples of contingencies:

p ∨ q
p → q
p ∧ q

6.4 Key Relationship

Important Relationship

Negation of a tautology = contradiction
Negation of a contradiction = tautology
If p is a tautology, ~p is a contradiction (and vice versa)
A statement p is a tautology iff ~p is a contradiction

PYQ — JEE Main 2021 (Aug 27 Shift 2)

Question: Which of the following is a tautology?

(a) p ∧ (p ∨ q)

(b) p ∨ (p ∧ q)

(c) p ∨ (p → q)

(d) p ↔ (p ∧ q)

Solution:

Check each option:

(c) p ∨ (p → q):

Recall p → q ≡ ~p ∨ q. So p ∨ (p → q) = p ∨ (~p ∨ q) = (p ∨ ~p) ∨ q = T ∨ q = T (always true!).

p	q	p→q	p∨(p→q)
T	T	T	T
T	F	F	T
F	T	T	T
F	F	T	T

All T → Tautology ✓

p \vee (p \to q) \text{ is a tautology}

Answer: (c) p ∨ (p → q)

7

Logical Equivalence

Two compound statements are logically equivalent if they have the same truth values for all possible combinations of their component statements. This concept is fundamental for simplifying logical expressions.

7.1 Definition

Most Repeated PYQ — Logical Equivalence

Two statements p and q are logically equivalent, denoted by p ≡ q (or p ⇔ q), if and only if:

p \leftrightarrow q \text{ is a tautology}

That is, p and q have the same truth value in every row of their truth table.

7.2 Important Logical Equivalences

Most Repeated PYQ — Equivalence Formulas

1. Implication: \(p \to q \equiv \sim p \vee q\)

2. Contrapositive: \(p \to q \equiv \sim q \to \sim p\)

3. Biconditional: \(p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)\)

4. Double Negation: \(\sim(\sim p) \equiv p\)

5. De Morgan 1: \(\sim(p \wedge q) \equiv \sim p \vee \sim q\)

6. De Morgan 2: \(\sim(p \vee q) \equiv \sim p \wedge \sim q\)

7. Negation of Implication: \(\sim(p \to q) \equiv p \wedge \sim q\)

8. Negation of Biconditional: \(\sim(p \leftrightarrow q) \equiv p \leftrightarrow \sim q \equiv (p \wedge \sim q) \vee (\sim p \wedge q)\)

9. Exportation: \((p \wedge q) \to r \equiv p \to (q \to r)\)

10. Idempotent: \(p \wedge p \equiv p\), \(p \vee p \equiv p\)

7.3 How to Verify Logical Equivalence

Method 1: Truth Table — Construct truth tables for both statements and verify that the final columns are identical.

Method 2: Algebraic Manipulation — Use known equivalences to transform one statement into the other.

PYQ — JEE Main 2022 (June 28 Shift 2)

Question: Which of the following is logically equivalent to ~(p → q)?

(a) p ∧ ~q

(b) ~p ∧ q

(c) p ∨ ~q

(d) ~p ∨ q

Solution:

Using p → q ≡ ~p ∨ q:

\sim(p \to q) \equiv \sim(\sim p \vee q)

By De Morgan's law: ~(A ∨ B) = ~A ∧ ~B, so:

\sim(\sim p \vee q) \equiv \sim(\sim p) \wedge \sim q \equiv p \wedge \sim q

\sim(p \to q) \equiv p \wedge \sim q

Answer: (a) p ∧ ~q

8

Algebra of Statements

Just like algebra of numbers, there is an algebra of statements governed by certain laws. These laws help us simplify and manipulate logical expressions, similar to how we simplify algebraic expressions.

8.1 Idempotent Laws

Most Repeated PYQ — Idempotent Laws

p \vee p \equiv p \quad \text{and} \quad p \wedge p \equiv p

8.2 Associative Laws

(p \vee q) \vee r \equiv p \vee (q \vee r)\] \[(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)

8.3 Commutative Laws

p \vee q \equiv q \vee p \quad \text{and} \quad p \wedge q \equiv q \wedge p

8.4 Distributive Laws

Most Repeated PYQ — Distributive Laws

p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)\] \[p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)

Note: AND distributes over OR, and OR distributes over AND (unlike arithmetic where only one direction works).

8.5 Identity Laws

Let T = tautology (always true), F = contradiction (always false). Then:

p ∨ F ≡ p (F is identity for ∨)
p ∧ T ≡ p (T is identity for ∧)
p ∨ T ≡ T
p ∧ F ≡ F

8.6 Complement Laws

p ∨ ~p ≡ T (Law of Excluded Middle)
p ∧ ~p ≡ F (Law of Contradiction)
~T ≡ F
~F ≡ T

8.7 Absorption Laws

p \vee (p \wedge q) \equiv p\] \[p \wedge (p \vee q) \equiv p

8.8 Involution Law (Double Negation)

\sim(\sim p) \equiv p

PYQ — JEE Main 2018 (Apr 16)

Question: Using algebra of statements, simplify: p ∨ (p ∧ q) ∨ (p ∧ ~q).

(a) p

(b) q

(c) p ∨ q

(d) p ∧ q

Solution:

p \vee (p \wedge q) \vee (p \wedge \sim q)

Group the last two terms:

\equiv p \vee [p \wedge (q \vee \sim q)]

Since q ∨ ~q = T (tautology):

\equiv p \vee (p \wedge T) \equiv p \vee p \equiv p

\text{Simplified: } p

Answer: (a) p

9

De Morgan's Laws

De Morgan's Laws are among the most important and frequently tested laws in mathematical reasoning. They describe how to distribute negation over conjunction and disjunction.

9.1 Statement of De Morgan's Laws

Most Repeated PYQ — De Morgan's Laws

First Law:

\sim(p \wedge q) \equiv \sim p \vee \sim q

"The negation of (p AND q) is (NOT p) OR (NOT q)"

Second Law:

\sim(p \vee q) \equiv \sim p \wedge \sim q

"The negation of (p OR q) is (NOT p) AND (NOT q)"

9.2 Generalization to n Statements

\sim(p_1 \wedge p_2 \wedge \cdots \wedge p_n) \equiv \sim p_1 \vee \sim p_2 \vee \cdots \vee \sim p_n\] \[\sim(p_1 \vee p_2 \vee \cdots \vee p_n) \equiv \sim p_1 \wedge \sim p_2 \wedge \cdots \wedge \sim p_n

9.3 Verification by Truth Table

p	q	~p	~q	p∧q	~(p∧q)	~p∨~q	p∨q	~(p∨q)	~p∧~q
T	T	F	F	T	F	F	T	F	F
T	F	F	T	F	T	T	T	F	F
F	T	T	F	F	T	T	T	F	F
F	F	T	T	F	T	T	F	T	T

Observe

~(p∧q) column = ~p∨~q column ✓

~(p∨q) column = ~p∧~q column ✓

9.4 English Example

Let p: "It is raining", q: "It is cold".

p ∧ q: "It is raining and it is cold."

~(p ∧ q): "It is not the case that it is raining and cold."

~p ∨ ~q: "Either it is not raining or it is not cold."

Both express the same meaning — De Morgan's Law in action.

PYQ — JEE Main 2021 (Jul 27 Shift 1)

Question: The negation of the statement "He is rich and he is happy" is:

(a) He is not rich or he is not happy

(b) He is not rich and he is not happy

(c) He is rich or he is happy

(d) He is rich and he is not happy

Solution:

Let p: "He is rich", q: "He is happy".

Statement: p ∧ q.

Negation: ~(p ∧ q) ≡ ~p ∨ ~q (De Morgan's First Law).

~p: "He is not rich", ~q: "He is not happy".

So ~p ∨ ~q: "He is not rich OR he is not happy."

\sim(p \wedge q) \equiv \sim p \vee \sim q

Answer: (a) He is not rich or he is not happy

10

Negation of Compound Statements

Finding the negation of compound statements is a high-yield topic for JEE. The key is to apply De Morgan's laws and other equivalences correctly.

10.1 Summary of Negation Formulas

Most Repeated PYQ — Negation Formulas

Statement	Negation
~p	p
p ∧ q	~p ∨ ~q
p ∨ q	~p ∧ ~q
p → q	p ∧ ~q
p ↔ q	(p ∧ ~q) ∨ (~p ∧ q)
p → (q ∧ r)	p ∧ (~q ∨ ~r)
p → (q ∨ r)	p ∧ (~q ∧ ~r)
(p ∧ q) → r	(p ∧ q) ∧ ~r

10.2 Detailed Derivations

1. Negation of p → q:

\sim(p \to q) \equiv \sim(\sim p \vee q) \equiv \sim(\sim p) \wedge \sim q \equiv p \wedge \sim q

2. Negation of p ↔ q:

p \leftrightarrow q \equiv (p \to q) \wedge (q \to p) \equiv (p \wedge \sim q) \vee (\sim p \wedge q)

So ~(p ↔ q) ≡ (p ↔ q)' which equals the negation, but easier: ~(p ↔ q) ≡ p ↔ ~q.

10.3 Negation with "AND", "OR" in English

Tips for Negating English Sentences

"AND" becomes "OR" when negated
"OR" becomes "AND" when negated
"For all" becomes "There exists...not"
"There exists" becomes "For all...not"
"If p then q" becomes "p and not q"

PYQ — JEE Main 2020 (Sep 5 Shift 1)

Question: The negation of the statement "If a number is divisible by 4, then it is divisible by 2" is:

(a) A number is divisible by 4 and it is not divisible by 2

(b) A number is not divisible by 4 or it is divisible by 2

(c) A number is not divisible by 4 and it is not divisible by 2

(d) If a number is not divisible by 4, then it is divisible by 2

Solution:

Let p: "A number is divisible by 4", q: "It is divisible by 2".

Statement: p → q.

Negation: ~(p → q) ≡ p ∧ ~q.

p ∧ ~q: "A number is divisible by 4 AND it is not divisible by 2."

\sim(p \to q) \equiv p \wedge \sim q

Answer: (a) A number is divisible by 4 and it is not divisible by 2

11

Quantifiers (Universal & Existential)

Quantifiers are words or phrases that indicate the quantity of elements for which a predicate holds. There are two main types: universal quantifier ("for all") and existential quantifier ("there exists").

11.1 Universal Quantifier (∀)

Most Repeated PYQ — Universal Quantifier

The universal quantifier, denoted by ∀ (read as "for all" or "for every"), asserts that a property holds for every element in a given set.

\forall x \in S, \; P(x)

Read as: "For all x in S, P(x) is true."

Example: ∀x ∈ ℝ, x² ≥ 0. ("For every real number x, x² is non-negative.")

English phrases: "for all", "for every", "for each", "given any".

11.2 Existential Quantifier (∃)

Most Repeated PYQ — Existential Quantifier

The existential quantifier, denoted by ∃ (read as "there exists" or "for some"), asserts that at least one element in a set satisfies the property.

\exists x \in S, \; P(x)

Read as: "There exists an x in S such that P(x) is true."

Example: ∃x ∈ ℝ, x² = 2. ("There exists a real number x such that x² = 2.") — x = √2.

English phrases: "there exists", "there is", "for some", "at least one".

11.3 Combining Quantifiers

Quantifiers can be combined in expressions:

∀x ∃y, P(x, y) — "For every x, there exists a y such that P(x,y)"
∃x ∀y, P(x, y) — "There exists an x such that for all y, P(x,y)"

Important: The order of quantifiers matters! ∀x ∃y P(x,y) ≠ ∃y ∀x P(x,y) in general.

11.4 Truth Values of Quantified Statements

∀x, P(x) is TRUE iff P(x) holds for every x.
∃x, P(x) is TRUE iff P(x) holds for at least one x.
∀x, P(x) is FALSE iff there exists some x for which P(x) is false.
∃x, P(x) is FALSE iff P(x) is false for all x.

PYQ — JEE Main 2019 (Jan 11 Shift 2)

Question: Which of the following is an existential quantifier statement?

(a) For every real number x, x² ≥ 0

(b) For all prime numbers p, p is odd

(c) There exists a real number x such that x² = 4

(d) Every natural number is positive

Solution:

Existential quantifier uses "there exists" (∃).

(a) "For every..." → Universal ∀

(b) "For all..." → Universal ∀

(c) "There exists..." → Existential ∃ ✓

(d) "Every..." → Universal ∀

\text{There exists a real number } x \text{ such that } x^2 = 4

Answer: (c) There exists a real number x such that x² = 4

12

Negation of Quantified Statements

Negating quantified statements follows simple but important rules. The negation of a universal statement is existential, and vice versa.

12.1 Rules for Negating Quantified Statements

Most Repeated PYQ — Negation of Quantifiers

Rule 1: Negation of Universal Statement

\sim(\forall x, P(x)) \equiv \exists x, \sim P(x)

"Not (for all x, P(x))" ≡ "There exists an x such that not P(x)"

Rule 2: Negation of Existential Statement

\sim(\exists x, P(x)) \equiv \forall x, \sim P(x)

"Not (there exists x, P(x))" ≡ "For all x, not P(x)"

12.2 Intuitive Understanding

Intuition

If "for all x, P(x)" is false, it means there is at least one x for which P(x) is false → "There exists x, ~P(x)".

If "there exists x, P(x)" is false, it means no x satisfies P(x) → "For all x, ~P(x)".

12.3 Examples

Example 1:

Statement: "For all x ∈ ℝ, x² > 0."

Negation: "There exists x ∈ ℝ such that x² ≤ 0."

Example 2:

Statement: "There exists a natural number n such that n² + 1 = 0."

Negation: "For all natural numbers n, n² + 1 ≠ 0."

12.4 Negation of Statements with Multiple Quantifiers

For statements with multiple quantifiers, negate each quantifier (changing ∀ to ∃ and vice versa) and negate the predicate:

\sim(\forall x, \exists y, P(x,y)) \equiv \exists x, \forall y, \sim P(x,y)\] \[\sim(\exists x, \forall y, P(x,y)) \equiv \forall x, \exists y, \sim P(x,y)

PYQ — JEE Main 2018 (Apr 9 Shift 2)

Question: The negation of the statement "Every prime number is odd" is:

(a) Every prime number is not odd

(b) No prime number is odd

(c) There exists a prime number which is not odd

(d) There exists a prime number which is odd

Solution:

"Every prime number is odd" is a universal statement: ∀ prime p, p is odd.

Negation: ~(∀p, odd(p)) ≡ ∃p, ~odd(p).

In words: "There exists a prime number which is not odd."

(Indeed, 2 is prime and even.)

\sim(\forall p, \text{odd}(p)) \equiv \exists p, \sim\text{odd}(p)

Answer: (c) There exists a prime number which is not odd

13

Duality Principle

The duality principle states that for any true statement in logic, there is a corresponding "dual" statement that is also true, obtained by interchanging certain connectives and constants.

13.1 Definition of Dual

Most Repeated PYQ — Duality

The dual of a compound statement is obtained by:

Replacing every ∧ (AND) with ∨ (OR)
Replacing every ∨ (OR) with ∧ (AND)
Replacing every T (True) with F (False)
Replacing every F (False) with T (True)
Keeping ~ (NOT), variables, and implications unchanged

13.2 Principle of Duality

Principle of Duality: If a statement S involving only ∧, ∨, ~, T, F is a tautology, then its dual S* is also a tautology.

Equivalently: If S ≡ T (tautology), then S* ≡ T (also tautology).

Or: If S₁ ≡ S₂, then S₁* ≡ S₂*.

13.3 Examples of Duals

Statement	Dual
p ∧ (q ∨ r)	p ∨ (q ∧ r)
(p ∧ T) ∨ (q ∧ F)	(p ∨ F) ∧ (q ∨ T)
p ∧ ~p ≡ F	p ∨ ~p ≡ T
~(p ∧ q) ≡ ~p ∨ ~q	~(p ∨ q) ≡ ~p ∧ ~q

PYQ — JEE Main 2017 (Apr 8)

Question: The dual of the statement (p ∧ q) ∨ ~r is:

(a) (p ∨ q) ∧ ~r

(b) (p ∨ q) ∧ r

(c) (p ∧ q) ∧ ~r

(d) (p ∨ q) ∨ ~r

Solution:

Replace ∧ with ∨ and ∨ with ∧. Leave ~r unchanged.

Original: (p ∧ q) ∨ ~r

Dual: (p ∨ q) ∧ ~r

(p \vee q) \wedge \sim r

Answer: (a) (p ∨ q) ∧ ~r

14

Open Sentences

An open sentence (or predicate) is a sentence that contains one or more variables and becomes a statement (with definite truth value) when the variables are assigned specific values.

14.1 Definition

Most Repeated PYQ — Open Sentences

An open sentence is a sentence of the form P(x) that contains a variable x such that:

When a specific value is substituted for x, P(x) becomes a statement (T or F).
Without substitution, P(x) has no truth value (so it's not a statement).

Example: "x + 2 = 5" is an open sentence. When x = 3, it becomes "3 + 2 = 5" (True). When x = 4, it becomes "4 + 2 = 5" (False).

14.2 Truth Set (Solution Set)

The truth set (or solution set) of an open sentence P(x) over a domain D is the set of all values of x in D for which P(x) is true.

\text{Truth set} = \{x \in D : P(x) \text{ is true}\}

Example: For P(x): "x² = 4" over domain ℝ, truth set = {-2, 2}.

14.3 Open Sentences with Multiple Variables

An open sentence may contain multiple variables: P(x, y), P(x, y, z), etc.

Example: "x + y = 10" is an open sentence in two variables. Substituting x = 3, y = 7 gives the statement "3 + 7 = 10" (True).

PYQ — JEE Main 2016 (Jul 3)

Question: Which of the following is an open sentence?

(a) 2 + 3 = 5

(b) The sun is a star

(c) x + 5 = 9

(d) π is irrational

Solution:

An open sentence contains a variable and has no definite truth value until the variable is assigned.

(a) "2 + 3 = 5" → True statement (no variable) ✗

(b) "The sun is a star" → True statement ✗

(c) "x + 5 = 9" → Contains variable x, truth depends on x → Open sentence ✓

(d) "π is irrational" → True statement ✗

x + 5 = 9 \text{ (open sentence)}

Answer: (c) x + 5 = 9

15

Validity of Arguments

An argument is a sequence of statements (premises) leading to a conclusion. The argument is valid if the conclusion logically follows from the premises — i.e., whenever all premises are true, the conclusion must also be true.

15.1 Definition of Valid Argument

Most Repeated PYQ — Validity

An argument with premises \(p_1, p_2, \ldots, p_n\) and conclusion \(q\) is valid if and only if:

(p_1 \wedge p_2 \wedge \cdots \wedge p_n) \to q \text{ is a tautology}

Equivalently: the conjunction of all premises logically implies the conclusion.

15.2 Standard Valid Argument Forms

1. Modus Ponens (Law of Detachment):

p \to q, \quad p \quad \vdash \quad q

If p → q is true and p is true, then q must be true.

2. Modus Tollens (Law of Contrapositive):

p \to q, \quad \sim q \quad \vdash \quad \sim p

If p → q is true and q is false, then p must be false.

3. Hypothetical Syllogism (Chain Rule):

p \to q, \quad q \to r \quad \vdash \quad p \to r

4. Disjunctive Syllogism:

p \vee q, \quad \sim p \quad \vdash \quad q

15.3 Invalid Argument Forms (Fallacies)

Common Fallacies

Fallacy of Affirming the Consequent: p → q, q ⊢ p (INVALID)
Fallacy of Denying the Antecedent: p → q, ~p ⊢ ~q (INVALID)

PYQ — JEE Main 2013

Question: Consider the argument: "If it rains, the ground is wet. It is raining. Therefore, the ground is wet." This argument is:

(a) Invalid

(b) Valid (Modus Ponens)

(c) Valid (Modus Tollens)

(d) Valid (Hypothetical Syllogism)

Solution:

Let p: "It rains", q: "The ground is wet".

Premise 1: p → q

Premise 2: p

Conclusion: q

This is Modus Ponens: (p → q) ∧ p → q is a tautology.

\text{Valid (Modus Ponens)}

Answer: (b) Valid (Modus Ponens)

16

Important Results & Summary

A consolidated list of all important results, equivalences, and formulas that frequently appear in JEE problems.

16.1 Master Formula Sheet

Most Repeated PYQ — Master Formulas

1. Implication: \(p \to q \equiv \sim p \vee q\)

2. Contrapositive: \(p \to q \equiv \sim q \to \sim p\)

3. Converse: \(q \to p\) (not equivalent to original)

4. Inverse: \(\sim p \to \sim q\) (not equivalent to original)

5. Biconditional: \(p \leftrightarrow q \equiv (p \to q) \wedge (q \to p)\)

6. De Morgan 1: \(\sim(p \wedge q) \equiv \sim p \vee \sim q\)

7. De Morgan 2: \(\sim(p \vee q) \equiv \sim p \wedge \sim q\)

8. Negation of implication: \(\sim(p \to q) \equiv p \wedge \sim q\)

9. Negation of biconditional: \(\sim(p \leftrightarrow q) \equiv p \leftrightarrow \sim q\)

10. Double negation: \(\sim(\sim p) \equiv p\)

11. Idempotent: \(p \wedge p \equiv p\), \(p \vee p \equiv p\)

12. Complement: \(p \vee \sim p \equiv T\), \(p \wedge \sim p \equiv F\)

13. Absorption: \(p \vee (p \wedge q) \equiv p\), \(p \wedge (p \vee q) \equiv p\)

14. Negation of universal: \(\sim(\forall x, P(x)) \equiv \exists x, \sim P(x)\)

15. Negation of existential: \(\sim(\exists x, P(x)) \equiv \forall x, \sim P(x)\)

16.2 Common Tautologies

\(p \vee \sim p\) (Law of Excluded Middle)
\(\sim(p \wedge \sim p)\) (Law of Non-Contradiction)
\(p \to p\)
\(p \to (q \to p)\)
\((p \to q) \vee (q \to p)\)
\(p \vee (p \to q)\)
\((p \to q) \vee (q \to r)\)
\((p \wedge (p \to q)) \to q\) (Modus Ponens)
\(((p \to q) \wedge (q \to r)) \to (p \to r)\) (Hypothetical Syllogism)

PYQ — JEE Main 2024 (Jan 27 Shift 1, if appeared) / JEE Main 2023 (Apr 11 Shift 1)

Question: The contrapositive of the statement "If two numbers are not equal, then their squares are not equal" is:

(a) If the squares of two numbers are equal, then the numbers are equal

(b) If the squares of two numbers are not equal, then the numbers are not equal

(c) If two numbers are equal, then their squares are equal

(d) If the squares of two numbers are equal, then the numbers are not equal

Solution:

Let p: "Two numbers are not equal", q: "Their squares are not equal".

Statement: p → q.

Contrapositive: ~q → ~p.

~q: "Their squares are equal" (not of "not equal").

~p: "The numbers are equal" (not of "not equal").

So contrapositive: "If the squares of two numbers are equal, then the numbers are equal."

\text{If squares are equal, then numbers are equal.}

Answer: (a) If the squares of two numbers are equal, then the numbers are equal

Previous Year Questions Analysis

JEE Main (Last 10 Years)

✓ Contrapositive of implication: 25%
✓ Negation of compound statements: 18%
✓ Tautology identification: 15%
✓ Logical equivalence: 12%
✓ De Morgan's laws: 10%
✓ Quantifiers & their negation: 8%
✓ Statement identification: 7%
✓ Connectives (AND, OR, NOT): 5%

Topic Distribution

✓ Converse, Inverse problems: 22%
✓ ~(p → q) ≡ p ∧ ~q: 18%
✓ Truth table based tautology: 14%
✓ p → q ≡ ~p ∨ q: 10%
✓ "For all"/"There exists" negation: 12%
✓ Duality: 8%
✓ Open sentences: 5%
✓ Validity of arguments: 6%
✓ Algebra of statements: 5%

Weightage Analysis

JEE Main (prior to 2024): 4 marks (1 question)

JEE Advanced: 4 marks (occasional)

Difficulty Level: Easy

Time Required: 3-4 hours practice

Download Complete PYQ PDF (2013-2023)

Practice Problem Set

Level 1: Basic (JEE Main Standard)

Which of the following is a statement? "What is your name?", "2+2=4", "x is an even number", "Close the door."
Construct the truth table for ~(p ∧ q).
Find the negation of "It is raining and it is cold."
Find the contrapositive of "If x is even, then x² is even."
Is p ∨ ~p a tautology? Justify.
State De Morgan's laws and verify with truth table.

Level 2: Intermediate

Find the dual of (p ∨ q) ∧ (r ∨ ~s).
Negate the statement "For all x, if x² > 0, then x ≠ 0."
Show that ~(p → q) ≡ p ∧ ~q using algebraic manipulation.
Check whether (p → q) ∧ (p → r) ≡ p → (q ∧ r).
Determine if the following argument is valid: "If today is Sunday, then I will rest. Today is Sunday. Therefore, I will rest."
Find the truth value of p → (~p ∨ q) when p = F, q = T.

Level 3: Advanced

Prove that (p → q) ∨ (q → r) is a tautology.
Show that ~[p ↔ q] ≡ (p ∧ ~q) ∨ (~p ∧ q).
Verify: ~(∀x ∃y, P(x,y)) ≡ ∃x ∀y, ~P(x,y).
Find the negation of "Every student who studies hard passes the exam."
Show that the dual of p ∧ ~p ≡ F is p ∨ ~p ≡ T.
Determine whether the argument "If it rains, the ground is wet. The ground is not wet. Therefore, it did not rain" is valid (Modus Tollens).

Get Solutions PDF with Step-by-Step Working

> FAQ

Frequently Asked Questions

What is a statement in mathematical reasoning?

A statement (or proposition) is a declarative sentence that is either true or false, but not both simultaneously. For example, "2 + 2 = 4" is a statement (true), "5 is a prime number" is a statement (true). Sentences like "How are you?" or "x + 1 = 5" (open, depends on x) are NOT statements.

What is the contrapositive of p → q?

The contrapositive of implication p → q is ~q → ~p. The contrapositive is logically equivalent to the original implication, i.e., p → q ≡ ~q → ~p. This is one of the most frequently asked concepts in JEE Mains Mathematical Reasoning.

What is a tautology?

A tautology is a compound statement that is always true, regardless of the truth values of its component statements. For example, p ∨ ~p (Law of Excluded Middle) and p → p are tautologies. The negation of a tautology is a contradiction (always false).

What are De Morgan's laws?

De Morgan's laws state: (1) ~(p ∧ q) ≡ ~p ∨ ~q (negation of conjunction is disjunction of negations), and (2) ~(p ∨ q) ≡ ~p ∧ ~q (negation of disjunction is conjunction of negations). These are fundamental laws used to simplify logical expressions.

What is the negation of p → q?

The negation of p → q is ~(p → q) ≡ p ∧ ~q. That is, the implication p → q is false only when p is true and q is false. So its negation is p ∧ ~q.

What are universal and existential quantifiers?

Universal quantifier (∀, "for all") states that a property holds for every element of a set. Existential quantifier (∃, "there exists") states that at least one element satisfies the property. Negation: ~(∀x, P(x)) ≡ ∃x, ~P(x) and ~(∃x, P(x)) ≡ ∀x, ~P(x).

What is logical equivalence?

Two compound statements are logically equivalent (denoted by ≡) if they have the same truth values for all possible truth values of their components. For example, p → q ≡ ~p ∨ q, and p ↔ q ≡ (p → q) ∧ (q → p).

What is the converse and inverse of p → q?

For implication p → q: (1) Converse is q → p, (2) Inverse is ~p → ~q, (3) Contrapositive is ~q → ~p. The converse and inverse are logically equivalent to each other, but neither is equivalent to the original implication.

How many questions come from Mathematical Reasoning in JEE?

Mathematical Reasoning typically contributed 1 question (4 marks) per JEE Main session before 2024. The chapter was removed from JEE Main syllabus from 2024 onwards, but the concepts remain relevant for JEE Advanced and foundational mathematics.

What is the duality principle?

The duality principle states that if a statement involving ∧, ∨, ~, T, F is a tautology, then its dual (obtained by replacing ∧ with ∨, ∨ with ∧, T with F, F with T) is also a tautology. For example, the dual of p ∧ ~p ≡ F is p ∨ ~p ≡ T.

Mathematical Reasoning notes, formulas & PYQs for JEE 2026

📑 Table of Contents

Mathematical Reasoning — An Overview

Statements & Propositions

1.1 Definition of a Statement

🔥 Most Repeated PYQ — Definition of Statement

1.2 Examples of Statements

1.3 Simple and Compound Statements

🔥 Most Repeated PYQ — Types of Statements

1.4 Components of a Compound Statement

📝 PYQ — JEE Main 2019 (Jan 12 Shift 1)

Truth Values & Truth Tables

2.1 Truth Values

🔥 Most Repeated PYQ — Truth Values

2.2 Construction of Truth Tables

🔥 Most Repeated PYQ — Number of Rows in Truth Table

2.3 Standard Pattern for Truth Table with Two Variables

💡 Memory Trick for Two Variables

📝 PYQ — JEE Main 2014 (Apr 19)

Logical Connectives (NOT, AND, OR)

3.1 Negation (NOT)

🔥 Most Repeated PYQ — Negation

3.2 Conjunction (AND)

🔥 Most Repeated PYQ — Conjunction

3.3 Disjunction (OR)

🔥 Most Repeated PYQ — Disjunction

3.4 Exclusive OR (XOR)

3.5 Summary Table of Connectives

📝 PYQ — JEE Main 2020 (Sep 2 Shift 1)

Conditional & Biconditional Statements

4.1 Conditional (Implication, IF-THEN)

🔥 Most Repeated PYQ — Implication

4.2 Important Equivalent Form of Implication

🔥 Most Repeated PYQ — p → q ≡ ~p ∨ q

4.3 Biconditional (IF AND ONLY IF)

🔥 Most Repeated PYQ — Biconditional

4.4 Order of Precedence of Connectives

📝 PYQ — JEE Main 2015 (Apr 10)

Converse, Inverse & Contrapositive

5.1 Definitions

🔥 Most Repeated PYQ — Converse, Inverse, Contrapositive

5.2 Verification by Truth Table

💡 Observe the Columns

5.3 Worked Examples

📝 PYQ — JEE Main 2023 (Apr 13 Shift 1)

Tautology, Contradiction & Contingency

6.1 Tautology

🔥 Most Repeated PYQ — Tautology

6.2 Contradiction

🔥 Most Repeated PYQ — Contradiction

6.3 Contingency

6.4 Key Relationship

💡 Important Relationship

📝 PYQ — JEE Main 2021 (Aug 27 Shift 2)

Logical Equivalence

7.1 Definition

🔥 Most Repeated PYQ — Logical Equivalence

7.2 Important Logical Equivalences

🔥 Most Repeated PYQ — Equivalence Formulas

7.3 How to Verify Logical Equivalence

📝 PYQ — JEE Main 2022 (June 28 Shift 2)

Algebra of Statements

8.1 Idempotent Laws

🔥 Most Repeated PYQ — Idempotent Laws

8.2 Associative Laws

8.3 Commutative Laws

8.4 Distributive Laws

🔥 Most Repeated PYQ — Distributive Laws

8.5 Identity Laws

8.6 Complement Laws

8.7 Absorption Laws

8.8 Involution Law (Double Negation)

📝 PYQ — JEE Main 2018 (Apr 16)

De Morgan's Laws

9.1 Statement of De Morgan's Laws

🔥 Most Repeated PYQ — De Morgan's Laws

9.2 Generalization to n Statements

9.3 Verification by Truth Table

💡 Observe

9.4 English Example

Table of Contents

Most Repeated PYQ — Definition of Statement

Most Repeated PYQ — Types of Statements

PYQ — JEE Main 2019 (Jan 12 Shift 1)

Most Repeated PYQ — Truth Values

Most Repeated PYQ — Number of Rows in Truth Table

Memory Trick for Two Variables

PYQ — JEE Main 2014 (Apr 19)

Most Repeated PYQ — Negation

Most Repeated PYQ — Conjunction

Most Repeated PYQ — Disjunction

PYQ — JEE Main 2020 (Sep 2 Shift 1)

Most Repeated PYQ — Implication

Most Repeated PYQ — p → q ≡ ~p ∨ q

Most Repeated PYQ — Biconditional

PYQ — JEE Main 2015 (Apr 10)

Most Repeated PYQ — Converse, Inverse, Contrapositive

Observe the Columns

PYQ — JEE Main 2023 (Apr 13 Shift 1)

Most Repeated PYQ — Tautology

Most Repeated PYQ — Contradiction

Important Relationship

PYQ — JEE Main 2021 (Aug 27 Shift 2)

Most Repeated PYQ — Logical Equivalence

Most Repeated PYQ — Equivalence Formulas

PYQ — JEE Main 2022 (June 28 Shift 2)

Most Repeated PYQ — Idempotent Laws

Most Repeated PYQ — Distributive Laws

PYQ — JEE Main 2018 (Apr 16)

Most Repeated PYQ — De Morgan's Laws

Observe

PYQ — JEE Main 2021 (Jul 27 Shift 1)

Most Repeated PYQ — Negation Formulas

Tips for Negating English Sentences

PYQ — JEE Main 2020 (Sep 5 Shift 1)

Most Repeated PYQ — Universal Quantifier

Most Repeated PYQ — Existential Quantifier

PYQ — JEE Main 2019 (Jan 11 Shift 2)

Most Repeated PYQ — Negation of Quantifiers

Intuition

PYQ — JEE Main 2018 (Apr 9 Shift 2)

Most Repeated PYQ — Duality

PYQ — JEE Main 2017 (Apr 8)

Most Repeated PYQ — Open Sentences

PYQ — JEE Main 2016 (Jul 3)

Most Repeated PYQ — Validity

Common Fallacies

PYQ — JEE Main 2013

Most Repeated PYQ — Master Formulas

PYQ — JEE Main 2024 (Jan 27 Shift 1, if appeared) / JEE Main 2023 (Apr 11 Shift 1)

Previous Year Questions Analysis

Practice Problem Set

Why This Chapter is Important

Must-Remember Formulas & Values