Mathematical Reasoning – JEE Main Maths Formula Sheet & Class 11 Notes | All Statements, Truth Tables, Connectives & Logical Formulas

Statements — Definition, Types, and What Is Not a Statement

Why Identifying Valid Mathematical Statements Is the First Step in All Mathematical Reasoning JEE Main Questions

Definition of a Statement (from PDF — Mathematical Reasoning):

A statement is an unambiguous declarative sentence that conveys a complete thought and is either true or false — but NOT both simultaneously. A sentence is called a mathematically acceptable statement when it satisfies this condition of being definitively true or definitively false.

Examples of valid Mathematical Reasoning statements:

"A.P.J. Abdul Kalam was President of India." — This is TRUE, and is a valid statement.

"2 + 2 = 5" — This is FALSE, but is a valid statement (statements can be false; they just cannot be ambiguous).

"All prime numbers are odd." — This is FALSE (2 is prime and even), but is a valid statement.

Statements are denoted by small letters p, q, r, s, … and written as: p : Ram is an engineer.

What is NOT a mathematically acceptable statement (from PDF — Mathematical Reasoning):

(1) Exclamatory sentences: "How sweet!" — Not a statement (neither true nor false).

(2) Imperative sentences (commands/requests): "Bring a glass of water." — Not a statement.

(3) Interrogative sentences (questions): "What are you doing?" — Not a statement.

(4) Sentences with variable time: "Yesterday was Sunday." / "It is raining today." — The truth value changes with time → not a statement in the mathematical sense.

(5) Sentences with pronouns not referring to specific persons: "He is an engineering graduate." — Ambiguous (who is "he"?) → not a statement.

(6) Sentences with variable places: "IIT is away from here." — "here" is ambiguous → not a statement.

(7) Sentences that may be true or false (ambiguous): "Mr. X will qualify JEE entrance exam." — Unknown truth value → not a mathematically acceptable statement.

Types of statements in Mathematical Reasoning:

Simple (atomic) statement: A statement that cannot be broken down further into sub-statements. Example: "p : 3 is a prime number."

Compound statement: A statement formed by combining two or more simple statements using connectives (and, or, if-then, iff). Example: "3 is a prime number and 4 is an even number." Download the Free PDF for Mathematical Reasoning for all statement examples for JEE Main.

Negation (~p) and Truth Table for Negation

Why Negation of a Statement Is the Most Fundamental Operation in Mathematical Reasoning

Negation (from PDF — Mathematical Reasoning):

The negation of a statement p is the denial of p. It is denoted by ~p (read: "not p"). The negation ~p is itself a statement, and it has the opposite truth value to p.

If p is TRUE, then ~p is FALSE. If p is FALSE, then ~p is TRUE.

Examples of Negation (from PDF — Mathematical Reasoning JEE Main):

p : "Everyone in India speaks Hindi." → ~p : "Everyone in India does not speak Hindi." (equivalently: "Not everyone in India speaks Hindi." / "There exists at least one person in India who does not speak Hindi.")

p : "2 is an even number." → ~p : "2 is not an even number."

p : "All triangles are equilateral." → ~p : "Not all triangles are equilateral." (i.e., there exists a triangle that is not equilateral).

Truth Table for Negation (from PDF — Mathematical Reasoning JEE Main):

The truth table for negation ~p:

p = T → ~p = F

p = F → ~p = T

In table form: | p | ~p | | T | F | | F | T |

Key rules for forming negation in Mathematical Reasoning:

(i) Negation of "All A are B" → "There exists at least one A that is not B."

(ii) Negation of "There exists an A that is B" → "All A are not B."

(iii) Negation of p→q (if p then q) → p ∧ ~q (p is true and q is false). This is the only case when p→q is false.

(iv) Negation of p↔q → (p∧~q) ∨ (~p∧q) — exactly one of p, q is true.

(v) De Morgan's Laws: ~(p∧q) = ~p ∨ ~q; ~(p∨q) = ~p ∧ ~q. Download the Free PDF for Mathematical Reasoning for all negation examples for JEE Main.

Connectives — Conjunction (AND), Disjunction (OR), Inclusive and Exclusive OR

Why Conjunction and Disjunction Truth Tables Are the Core Building Blocks of Mathematical Reasoning

Connectives in Mathematical Reasoning (from PDF): Connecting words used to form compound statements. The main connectives are "And" (∧), "Or" (∨), "If-then" (→), and "If and only if" (↔). The words "There exists" (∃) and "For all" (∀) are called quantifiers.

Conjunction — "And" — ∧ (from PDF — Mathematical Reasoning JEE Main):

A compound statement p ∧ q (read: "p and q") is formed by connecting p and q with "and".

Rule for Conjunction:

(i) p ∧ q is TRUE if and only if BOTH p and q are TRUE.

(ii) p ∧ q is FALSE if ANY ONE component statement is FALSE.

Truth Table for Conjunction p ∧ q (from PDF):

p=T, q=T → p∧q = T

p=T, q=F → p∧q = F

p=F, q=T → p∧q = F

p=F, q=F → p∧q = F

Important note from PDF: A statement with "And" is not always a compound statement. Example: "Bread and Butter is good for health" — here "and" connects two words within one thought, not two separate mathematical statements.

Disjunction — "Or" — ∨ (from PDF — Mathematical Reasoning JEE Main):

A compound statement p ∨ q (read: "p or q") is formed by connecting p and q with "or".

Rule for Disjunction:

(i) p ∨ q is TRUE when at least one of p or q is TRUE (one is true, or both are true).

(ii) p ∨ q is FALSE only when BOTH p and q are FALSE.

Truth Table for Disjunction p ∨ q (from PDF):

p=T, q=T → p∨q = T

p=T, q=F → p∨q = T

p=F, q=T → p∨q = T

p=F, q=F → p∨q = F

Inclusive OR vs Exclusive OR (from PDF — Mathematical Reasoning):

Inclusive OR: "p or q" means p is true, or q is true, or BOTH can be true. This is the standard ∨ (disjunction) used in mathematics. Example: "A student who has taken biology OR chemistry can apply for M.Sc microbiology." — A student who has taken both also qualifies.

Exclusive OR (↓ or XOR): "p or q" means EXACTLY ONE of p or q is true — both being true is excluded. Denoted p ↓ q. Example: "An ice cream or a pepsi is available with a Thali" — you can have one, not both. Truth table for exclusive OR: T,T→F; T,F→T; F,T→T; F,F→F.

All Mathematical Reasoning symbols (from PDF):

~p : Negation of p ("not p"). p∧q : Conjunction ("p and q"). p∨q : Disjunction ("p or q"). p↓q : Exclusive disjunction ("only one of p and q"). p↓↓q (or p↓q joint denial) : "Neither p nor q". p→q : Conditional / Implication ("if p then q"). p↔q : Biconditional / Equivalence ("p if and only if q"). Download the Free PDF for Mathematical Reasoning for all connective truth table examples for JEE Main.

Implications — If-Then (p→q), Converse, Contrapositive, and Biconditional

Why Implications, Contrapositive, and Converse Are the Most-Tested Mathematical Reasoning Topics in JEE Main

Implication / Conditional — "If p then q" — p→q (from PDF — Mathematical Reasoning JEE Main):

The statement "If p, then q" (written p→q) means: whenever p is true, q must also be true. It is called a conditional statement or implication. Read as: "p implies q".

Truth Table for Implication p→q (from PDF — Mathematical Reasoning):

p=T, q=T → p→q = T

p=T, q=F → p→q = F (only false case)

p=F, q=T → p→q = T

p=F, q=F → p→q = T

Key insight from PDF: p→q is FALSE only when p is TRUE and q is FALSE. When p is false, p→q is true regardless of q — the implication says nothing about q when p is false.

All equivalent forms of p→q (from PDF — Mathematical Reasoning JEE Main):

Let p : "a number is a multiple of 8" and q : "a number is a multiple of 4". Then p→q can be expressed as:

(i) p implies q — denoted p→q.

(ii) p is a sufficient condition for q — "being a multiple of 8 is sufficient to conclude it is a multiple of 4."

(iii) p only if q — "a number is a multiple of 8 ONLY IF it is a multiple of 4."

(iv) q is a necessary condition for p — "being a multiple of 4 is NECESSARY for a number to be a multiple of 8."

(v) ~q implies ~p — "if not a multiple of 4, then not a multiple of 8." [This is the CONTRAPOSITIVE of p→q.]

Contrapositive of p→q (from PDF — Mathematical Reasoning JEE Main):

The contrapositive of "if p then q" is "if ~q then ~p" (i.e., ~q → ~p).

Example: p→q: "If the physical environment changes, then the biological environment changes."

Contrapositive (~q→~p): "If the biological environment does NOT change, then the physical environment does NOT change."

Critical fact: A statement and its contrapositive are logically equivalent — they have exactly the same truth table. p→q ≡ ~q→~p.

Converse of p→q (from PDF — Mathematical Reasoning JEE Main):

The converse of "if p then q" is "if q then p" (i.e., q→p).

Example: p→q: "If a number is divisible by 20, then it is divisible by 10."

Converse (q→p): "If a number is divisible by 10, then it is divisible by 20." [This is FALSE — 30 is divisible by 10 but not 20.]

Critical fact: The converse of a true statement may be FALSE. Converse is NOT logically equivalent to the original statement.

Biconditional — "p if and only if q" — p↔q (from PDF — Mathematical Reasoning JEE Main):

The statement p↔q (read: "p if and only if q", abbreviated "p iff q") is true when p and q have the SAME truth value (both true or both false).

Truth Table for Biconditional p↔q (from PDF):

p=T, q=T → p↔q = T

p=T, q=F → p↔q = F

p=F, q=T → p↔q = F

p=F, q=F → p↔q = T

All equivalent forms of p↔q (from PDF — Mathematical Reasoning):

(i) p if and only if q. (ii) q if and only if p. (iii) p is a necessary AND sufficient condition for q (and vice versa). (iv) p→q AND q→p (biconditional = implication in both directions). Download the Free PDF for Mathematical Reasoning for all implication, contrapositive, converse, and iff examples for JEE Main.

Quantifiers — "There Exists" (∃) and "For All" (∀)

Why Quantifiers and Their Negations Are Directly Tested in Mathematical Reasoning JEE Main Questions

Quantifiers in Mathematical Reasoning (from PDF): Quantifiers are special phrases used to indicate the range over which a statement applies. The two main quantifiers in Mathematical Reasoning are:

"There exists" — ∃ (from PDF — Mathematical Reasoning):

"There exists" means "at least one". Example from PDF: "There exists a rectangle whose sides are equal." This means there is AT LEAST ONE rectangle whose all sides are equal (i.e., a square).

In symbolic form: ∃x such that P(x) is true — at least one element x in the domain makes P(x) true.

"For all" / "For every" — ∀ (from PDF — Mathematical Reasoning):

"For all" (or "for every") means the statement holds for EVERY element in the domain. Example from PDF: "For every prime number p, √p is an irrational number." This means if S is the set of all prime numbers, then for ALL members p of S, √p is irrational.

In symbolic form: ∀x, P(x) is true — every element x in the domain makes P(x) true.

Negation of quantified statements (Mathematical Reasoning — JEE Main):

Negation of "∀x, P(x)" → "∃x such that ~P(x)" [There exists at least one x for which P(x) is false]

Negation of "∃x, P(x)" → "∀x, ~P(x)" [For all x, P(x) is false]

Example: Negation of "All triangles are equilateral" → "There exists a triangle that is not equilateral." (Negation of a ∀-statement is an ∃-statement.)

Example: Negation of "There exists a real number x such that x² = –1" → "For all real numbers x, x² ≠ –1." Download the Free PDF for Mathematical Reasoning for all quantifier examples for JEE Main.

All 6 Rules for Validating Statements in Mathematical Reasoning

Why All 6 Validation Methods Are Complete-in-Themselves Mathematical Reasoning JEE Main Formulas

Overview (from PDF — Mathematical Reasoning): To check whether a statement is true, we must understand: what the statement means; what it means for the statement to be true; and what it means for the statement to be false. The method of validation depends on the connectives, implications, and quantifiers present in the statement.

Rule 1 — Validating "p AND q" (from PDF — Mathematical Reasoning JEE Main):

To show that the compound statement "p ∧ q" is TRUE:

Case 1: Show that statement p is TRUE.

Case 2: Show that statement q is TRUE.

Both must be demonstrated independently. The conjunction is true only when both components are true.

Rule 2 — Validating "p OR q" (from PDF — Mathematical Reasoning JEE Main):

To show that the compound statement "p ∨ q" is TRUE, either:

Case 1: Assume p is FALSE — then show that q must be TRUE.

Case 2: Assume q is FALSE — then show that p must be TRUE.

It is enough to show one case — if one component can be shown true under the assumption the other is false, the disjunction holds.

Rule 3 — Validating "if p then q" (from PDF — Mathematical Reasoning JEE Main):

Two methods to prove p→q is TRUE:

Case 1: Direct Method — Assume p is TRUE, then show that q must be TRUE.

Case 2: Contrapositive Method — Assume q is FALSE (~q is true), then show that p must be FALSE (~p is true). Since ~q→~p is logically equivalent to p→q, proving the contrapositive proves the original implication.

Rule 4 — Validating "p if and only if q" (from PDF — Mathematical Reasoning JEE Main):

To prove p↔q is TRUE, prove BOTH directions:

(i) If p is TRUE, then q is TRUE (prove p→q).

(ii) If q is TRUE, then p is TRUE (prove q→p).

The biconditional holds only when both forward and backward implications hold.

Rule 5 — Proof by Contradiction (from PDF — Mathematical Reasoning JEE Main):

To prove that a statement p is TRUE using contradiction:

Step 1: Assume p is NOT true, i.e., assume ~p is true.

Step 2: From the assumption ~p, derive a logical consequence.

Step 3: Show that this consequence leads to a contradiction (contradicts a known true fact, or contradicts the assumption itself).

Step 4: Since the assumption ~p leads to a contradiction, ~p must be FALSE → therefore p is TRUE.

This is one of the most powerful methods in Mathematical Reasoning and all of mathematics.

Rule 6 — Proof by Counter Example (from PDF — Mathematical Reasoning JEE Main):

To show that a general statement is FALSE, it is sufficient to find ONE counter example — one specific case where the statement fails.

Example from PDF: Statement: "If n is an odd integer, then n is prime." To disprove: n = 9 is odd, but 9 = 3×3 is NOT prime. This single counter example is sufficient to prove the statement is FALSE.

Key: a statement of the form "For all x, P(x)" is FALSE if even ONE x can be found for which P(x) is false. Download the Free PDF for Mathematical Reasoning for all validation rule examples for JEE Main.

Tautology, Contradiction, Logical Equivalence, and De Morgan's Laws

Why Tautology, Contradiction, and De Morgan's Laws Complete the Mathematical Reasoning JEE Main Formula Set

Tautology (Mathematical Reasoning — JEE Main): A compound statement is called a tautology if it is TRUE for all possible combinations of truth values of its component statements. A tautology's truth table column consists entirely of T.

Example: p ∨ ~p is always TRUE (either p is true, or ~p is true — one must be). Truth table: p=T→T∨F=T; p=F→F∨T=T. Always T → tautology.

Example: p→p is always TRUE (a statement implies itself). Example: (p→q) ↔ (~q→~p) is always TRUE — shows p→q and its contrapositive are logically equivalent.

Contradiction / Fallacy (Mathematical Reasoning — JEE Main): A compound statement is called a contradiction (or fallacy) if it is FALSE for all possible combinations of truth values. Its truth table column consists entirely of F.

Example: p ∧ ~p is always FALSE (a statement cannot be both true and not true simultaneously). Truth table: p=T→T∧F=F; p=F→F∧T=F. Always F → contradiction.

Logical Equivalence (Mathematical Reasoning — JEE Main): Two statements A and B are logically equivalent (written A ≡ B) if they have IDENTICAL truth tables — i.e., they are true and false for exactly the same combinations of truth values of the component statements.

Key logical equivalences in Mathematical Reasoning:

p→q ≡ ~q→~p (implication ≡ contrapositive)

p→q ≡ ~p ∨ q (implication in terms of disjunction)

~(p∧q) ≡ ~p ∨ ~q (De Morgan's law 1)

~(p∨q) ≡ ~p ∧ ~q (De Morgan's law 2)

p↔q ≡ (p→q) ∧ (q→p)

De Morgan's Laws for Mathematical Reasoning (JEE Main):

Law 1: ~(p ∧ q) ≡ ~p ∨ ~q

The negation of "p AND q" is "NOT p OR NOT q".

Example: Negation of "It is raining AND it is cold" → "It is not raining OR it is not cold."

Law 2: ~(p ∨ q) ≡ ~p ∧ ~q

The negation of "p OR q" is "NOT p AND NOT q".

Example: Negation of "It is raining OR it is cold" → "It is not raining AND it is not cold."

Inductive vs Deductive Reasoning (from PDF — Mathematical Reasoning):

Inductive reasoning: Reasoning from specific cases to general conclusions (used in Mathematical Induction). Observing specific examples and forming a general rule.

Deductive reasoning: Reasoning from general principles to specific conclusions (the basis of all Mathematical Reasoning in this chapter). Starting from axioms/known truths and deriving specific conclusions using logic. Download the Free PDF for Mathematical Reasoning for all tautology, contradiction, and De Morgan's law examples for JEE Main.