Module 7 · JEE Mathematics
Mathematical Reasoning — The Grammar of Logic
Statements, truth tables, connectives, tautologies — the only JEE Main topic that's 100% logic, 0% calculation.
📝 Statements
🔗 Connectives
✅ Tautologies
🧠 Arguments
🎯 Practice
📝 Part 1: Statements — The Building Blocks of Logic
🔗 Part 2: Logical Connectives — AND, OR, NOT, IF-THEN, IFF
✅ Part 3: Tautology, Contradiction & Contingency
The last column is ALL T. Therefore, (p → q) ∨ (q → p) is a tautology. Intuitively: no matter what p and q are, at least one direction of the implication is always true!
🧠 Part 4: Valid Arguments — Syllogisms and Proof Patterns
⚠️ Common Mistakes (Don't Fall For These!)
🧠 Mathematical Reasoning Toolkit
🔥 Advanced Worked Examples (JEE Level)
🎯 Practice Zone: Test Your Understanding
A Statement Is a Sentence That Is Either True or False (But Not Both)
Mathematical reasoning starts with statements — declarative sentences that have a definite truth value (T or F).
📝 What Is a Statement?
"2 + 2 = 4" ✅ True. "2 + 2 = 5" ✅ False. "Hello!" ❌ Not a statement.
Statement: A sentence that is either true or false, but not both.
Examples of statements:
• "New Delhi is the capital of India." → True (T)
• "5 is greater than 7." → False (F)
• "Every even number is divisible by 2." → True (T)
Not statements:
• "Hello!" (exclamation — no truth value)
• "Is it raining?" (question — no truth value)
• "x + 2 = 5" (open sentence — depends on x, not a statement unless quantified)
• "This statement is false." (paradox — can't be consistently assigned T or F)
Examples of statements:
• "New Delhi is the capital of India." → True (T)
• "5 is greater than 7." → False (F)
• "Every even number is divisible by 2." → True (T)
Not statements:
• "Hello!" (exclamation — no truth value)
• "Is it raining?" (question — no truth value)
• "x + 2 = 5" (open sentence — depends on x, not a statement unless quantified)
• "This statement is false." (paradox — can't be consistently assigned T or F)
💡 Statements = declarative sentences with a truth value. Questions, exclamations, commands, and paradoxes are NOT statements.
Compound Statements — Connecting Simple Statements
Simple statements like "It is raining" can be combined with logical connectives to form compound statements like "It is raining AND it is cold."
📊 Simple vs Compound Statements
Simple statement: One atomic claim. Compound statement: Multiple simple statements linked by logical connectives. The truth value of a compound depends on the truth values of its parts and the connective.
Five Connectives. Five Truth Tables. That's All There Is.
Each logical connective has a precise meaning defined by its truth table — a table that shows the truth value of the compound for every combination of input truth values.
Negation (¬) — The NOT Operator
¬p simply flips the truth value. If p is true, ¬p is false. If p is false, ¬p is true.
📊 Truth Tables — All 5 Connectives
The five truth tables: ¬p flips. p∧q is T only when both are T. p∨q is T when at least one is T. p→q (implication) is F only when T implies F. p↔q is T when both have the same truth value.
The Tricky One: Implication (p → q)
The implication "if p then q" is the most misunderstood connective. It's false ONLY when p is true and q is false. In all other cases — even when p is false — the implication is considered TRUE.
→ The Implication
"If it rains, the ground gets wet." p → q is TRUE even when it doesn't rain!
Why is p → q true when p is false? Because the implication only promises: IF p happens, THEN q happens. If p doesn't happen, the promise isn't broken!
Example: "If you score 100%, you get an A+."
• You score 100% AND get A+ → Promise kept. (T → T = T) ✅
• You score 100% BUT don't get A+ → Promise broken. (T → F = F) ❌
• You DON'T score 100% AND get or don't get A+ → Promise wasn't tested. (F → T/F = T) ✅
Key insight: An implication is a promise. It's only broken when the condition is met but the result doesn't happen.
Example: "If you score 100%, you get an A+."
• You score 100% AND get A+ → Promise kept. (T → T = T) ✅
• You score 100% BUT don't get A+ → Promise broken. (T → F = F) ❌
• You DON'T score 100% AND get or don't get A+ → Promise wasn't tested. (F → T/F = T) ✅
Key insight: An implication is a promise. It's only broken when the condition is met but the result doesn't happen.
💡 p → q is false ONLY when p is true and q is false. "False implies anything" is vacuously true.
Contrapositive, Converse, Inverse
For any implication p → q, we can form three related statements. The contrapositive is logically equivalent to the original.
📊 Implication and Its Relatives
Original ⇔ Contrapositive (logically equivalent). Converse and Inverse are not equivalent to the original, but they are equivalent to each other. The contrapositive is the most powerful tool for proving implications.
Tautology = Always True. Contradiction = Always False. Contingency = Sometimes True.
These classifications come from the truth table of a compound statement.
✅ Tautology
A Statement That Is TRUE in Every Possible Case — Like "p OR ¬p"
Tautology: The last column of the truth table is ALL TRUE.
Contradiction: The last column of the truth table is ALL FALSE.
Contingency: The truth table has BOTH true and false entries.
Classic tautologies:
• p ∨ ¬p (Law of Excluded Middle — "Either it is raining or it is not raining")
• ¬(p ∧ ¬p) (Law of Non-Contradiction — "It can't be both raining and not raining")
• (p ∧ q) → p (Simplification — "If p AND q are true, then p is true")
Method to check: Construct the truth table. If ALL entries in the final column are T → tautology. ALL F → contradiction. Mixed → contingency.
Contradiction: The last column of the truth table is ALL FALSE.
Contingency: The truth table has BOTH true and false entries.
Classic tautologies:
• p ∨ ¬p (Law of Excluded Middle — "Either it is raining or it is not raining")
• ¬(p ∧ ¬p) (Law of Non-Contradiction — "It can't be both raining and not raining")
• (p ∧ q) → p (Simplification — "If p AND q are true, then p is true")
Method to check: Construct the truth table. If ALL entries in the final column are T → tautology. ALL F → contradiction. Mixed → contingency.
💡 Tautology = always T (like p ∨ ¬p). Contradiction = always F (like p ∧ ¬p). Contingency = sometimes T, sometimes F.
Worked Example: Checking a Tautology
Show that (p → q) ∨ (q → p) is a tautology.
1
Construct the truth table
▶
| p | q | p→q | q→p | (p→q) ∨ (q→p) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | T | F | T |
| F | F | T | T | T |
An Argument Is Valid If the Conclusion Follows from the Premises
In mathematical reasoning, an argument consists of premises (given statements) and a conclusion. The argument is valid if whenever ALL premises are true, the conclusion must also be true.
🧠 Valid Arguments
Modus Ponens, Modus Tollens, and the Syllogism
Modus Ponens (Mode of Affirming):
Premise 1: p → q
Premise 2: p
Conclusion: ∴ q
"If it rains, the ground gets wet. It rains. Therefore, the ground gets wet." ✅
Modus Tollens (Mode of Denying):
Premise 1: p → q
Premise 2: ¬q
Conclusion: ∴ ¬p
"If it rains, the ground gets wet. The ground is NOT wet. Therefore, it did NOT rain." ✅
Hypothetical Syllogism:
Premise 1: p → q
Premise 2: q → r
Conclusion: ∴ p → r
"Chain of implications: if p then q, if q then r, so if p then r." ✅
Fallacies (invalid arguments to watch for):
• Fallacy of the converse: p → q and q does NOT imply p ❌
• Fallacy of the inverse: p → q and ¬p does NOT imply ¬q ❌
Premise 1: p → q
Premise 2: p
Conclusion: ∴ q
"If it rains, the ground gets wet. It rains. Therefore, the ground gets wet." ✅
Modus Tollens (Mode of Denying):
Premise 1: p → q
Premise 2: ¬q
Conclusion: ∴ ¬p
"If it rains, the ground gets wet. The ground is NOT wet. Therefore, it did NOT rain." ✅
Hypothetical Syllogism:
Premise 1: p → q
Premise 2: q → r
Conclusion: ∴ p → r
"Chain of implications: if p then q, if q then r, so if p then r." ✅
Fallacies (invalid arguments to watch for):
• Fallacy of the converse: p → q and q does NOT imply p ❌
• Fallacy of the inverse: p → q and ¬p does NOT imply ¬q ❌
💡 Valid argument = when premises are true, conclusion MUST be true. Invalid = fallacy. Modus ponens and tollens are the two workhorses of logical proof.
Worked Example: Checking Argument Validity
Check if the following argument is valid: "If I study, I pass. I pass. Therefore, I studied."
1
Translate to symbols and check
▶
Let p = "I study", q = "I pass."
Premise 1: p → q
Premise 2: q
Conclusion: ∴ p
This is the fallacy of the converse! From p → q and q, we CANNOT conclude p.
Why? What if I passed because the test was easy? I could pass without studying. The premises don't rule out other ways to pass. So the argument is INVALID.
Truth table check: Is there a case where premises are T but conclusion is F?
p = F, q = T: p→q = T (F→T = T), q = T, but p = F. Yes! This row shows the argument is invalid.
Premise 1: p → q
Premise 2: q
Conclusion: ∴ p
This is the fallacy of the converse! From p → q and q, we CANNOT conclude p.
Why? What if I passed because the test was easy? I could pass without studying. The premises don't rule out other ways to pass. So the argument is INVALID.
Truth table check: Is there a case where premises are T but conclusion is F?
p = F, q = T: p→q = T (F→T = T), q = T, but p = F. Yes! This row shows the argument is invalid.
🌟 Argument Validity Cheat Sheet
Valid: Modus Ponens (p→q, p ⊢ q) | Modus Tollens (p→q, ¬q ⊢ ¬p) | Syllogism (p→q, q→r ⊢ p→r)
Invalid: Converse fallacy (p→q, q ⊢ p) ❌ | Inverse fallacy (p→q, ¬p ⊢ ¬q) ❌
These Traps Trip Up Even Top Students
⚠️ Mistake 1
Confusing OR (∨) in Logic with "Either...Or" in English
Wrong: p ∨ q means "p or q but not both" ❌
Right: In logic, p ∨ q is inclusive OR — at least one is true (could be both).
The "exclusive or" (either p or q but not both) is a different connective called XOR (⊻).
Example: "I will have tea OR coffee" in everyday English usually means one OR the other but not both. But in logic, p ∨ q includes the possibility of both! If you want exactly one, use XOR.
Right: In logic, p ∨ q is inclusive OR — at least one is true (could be both).
The "exclusive or" (either p or q but not both) is a different connective called XOR (⊻).
Example: "I will have tea OR coffee" in everyday English usually means one OR the other but not both. But in logic, p ∨ q includes the possibility of both! If you want exactly one, use XOR.
⚠️ ∨ = inclusive OR (at least one). English "or" is often exclusive. In JEE, ∨ is ALWAYS inclusive unless specified as "either...or...but not both."
⚠️ Mistake 2
Thinking "If p then q" Means the Same as "p Causes q"
Wrong: p → q means p causes q ❌
Right: p → q is about truth values, not causation. "If 2+2=4, then Paris is the capital of France" is TRUE (both true). "If 2+2=5, then I'm the Pope" is TRUE (false → anything is true).
The implication only says: "It is NOT the case that p is true and q is false simultaneously." It has nothing to do with causality, time, or relevance!
Right: p → q is about truth values, not causation. "If 2+2=4, then Paris is the capital of France" is TRUE (both true). "If 2+2=5, then I'm the Pope" is TRUE (false → anything is true).
The implication only says: "It is NOT the case that p is true and q is false simultaneously." It has nothing to do with causality, time, or relevance!
⚠️ p → q is a truth-functional connective, not a causal statement. It's only about combinations of truth values, not about whether p actually "causes" q.
⚠️ Mistake 3
Confusing the Converse with the Contrapositive
Wrong: p → q ≡ q → p ❌
Right: p → q ≡ ¬q → ¬p (contrapositive). The converse q → p is NOT equivalent!
Example: "If it's a dog, it's an animal" (p → q) is true.
• Contrapositive: "If it's NOT an animal, it's NOT a dog." (¬q → ¬p) ✅ Also true.
• Converse: "If it's an animal, it's a dog." (q → p) ❌ False! (Could be a cat.)
Learning to distinguish the contrapositive (equivalent) from the converse (not equivalent) is crucial for JEE.
Right: p → q ≡ ¬q → ¬p (contrapositive). The converse q → p is NOT equivalent!
Example: "If it's a dog, it's an animal" (p → q) is true.
• Contrapositive: "If it's NOT an animal, it's NOT a dog." (¬q → ¬p) ✅ Also true.
• Converse: "If it's an animal, it's a dog." (q → p) ❌ False! (Could be a cat.)
Learning to distinguish the contrapositive (equivalent) from the converse (not equivalent) is crucial for JEE.
⚠️ Contrapositive = ¬q → ¬p (equivalent to original). Converse = q → p (NOT equivalent). Don't swap them!
4 Strategies That Crack Almost Any Problem
🧠 Strategy 1
"Build the Truth Table" — The Universal Method
When in doubt, build a truth table. For n statements, you need 2ⁿ rows. For JEE, n is almost always 2 (4 rows) or sometimes 3 (8 rows).
Steps:
1. List all combinations of T/F for the atomic statements.
2. Compute intermediate columns for sub-statements.
3. Compute the final column.
4. Analyze: all T = tautology, all F = contradiction, mixed = contingency.
Pro tip: For tautology checking, you can also use the method of contradiction — assume the statement is false and check if that leads to a contradiction.
Steps:
1. List all combinations of T/F for the atomic statements.
2. Compute intermediate columns for sub-statements.
3. Compute the final column.
4. Analyze: all T = tautology, all F = contradiction, mixed = contingency.
Pro tip: For tautology checking, you can also use the method of contradiction — assume the statement is false and check if that leads to a contradiction.
💡 Truth tables are the universal tool. For 2 statements (4 rows), you can always build the truth table within a minute.
🧠 Strategy 2
"Use Known Equivalences" Instead of Full Truth Tables
For speed, memorize these logical equivalences:
• De Morgan's Laws: ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q
• Implication: p → q ≡ ¬p ∨ q (very useful for simplifying!)
• Contrapositive: p → q ≡ ¬q → ¬p
• Double negation: ¬(¬p) ≡ p
• Distributive: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
These let you transform compound statements into simpler equivalent forms without building truth tables.
• De Morgan's Laws: ¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q
• Implication: p → q ≡ ¬p ∨ q (very useful for simplifying!)
• Contrapositive: p → q ≡ ¬q → ¬p
• Double negation: ¬(¬p) ≡ p
• Distributive: p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
These let you transform compound statements into simpler equivalent forms without building truth tables.
💡 p → q ≡ ¬p ∨ q is the single most useful equivalence. It rewrites implication as OR, which is often easier to work with.
🧠 Strategy 3
"For Valid Arguments, Look for the Counterexample"
To check if an argument is valid, try to find a counterexample — a situation where all premises are true but the conclusion is false.
If you can find one, the argument is invalid. If no such situation exists, it's valid.
Example: "If it's a square (p), it has 4 sides (q). This shape has 4 sides. Therefore it's a square."
Counterexample: A rectangle has 4 sides but isn't necessarily a square. Premises T, conclusion F → invalid.
If you can find one, the argument is invalid. If no such situation exists, it's valid.
Example: "If it's a square (p), it has 4 sides (q). This shape has 4 sides. Therefore it's a square."
Counterexample: A rectangle has 4 sides but isn't necessarily a square. Premises T, conclusion F → invalid.
💡 Invalid argument = there exists at least one case where premises are T but conclusion is F. Find that case, and you've disproven the argument.
🧠 Strategy 4
"Quantified Statements — Watch the Order"
For JEE Main, you may encounter statements with quantifiers:
• ∀ (for all): "∀x > 0, x² > 0" → true
• ∃ (there exists): "∃x > 0, x² = 2" → true (x = √2)
Negation of quantifiers:
• ¬(∀x P(x)) ≡ ∃x ¬P(x) — "Not all birds fly" ≡ "There exists a bird that doesn't fly"
• ¬(∃x P(x)) ≡ ∀x ¬P(x) — "There is no unicorn" ≡ "Everything is not a unicorn"
Key trick: To negate a quantified statement: flip ∀ ↔ ∃ and negate the predicate.
• ∀ (for all): "∀x > 0, x² > 0" → true
• ∃ (there exists): "∃x > 0, x² = 2" → true (x = √2)
Negation of quantifiers:
• ¬(∀x P(x)) ≡ ∃x ¬P(x) — "Not all birds fly" ≡ "There exists a bird that doesn't fly"
• ¬(∃x P(x)) ≡ ∀x ¬P(x) — "There is no unicorn" ≡ "Everything is not a unicorn"
Key trick: To negate a quantified statement: flip ∀ ↔ ∃ and negate the predicate.
💡 ¬(∀x P(x)) = ∃x ¬P(x). ¬(∃x P(x)) = ∀x ¬P(x). Flip the quantifier and negate the predicate.
Multi-Step Problems That Combine Multiple Concepts
Example 1: Tautology Check Using Equivalence
Show that (p → q) ∧ (p → ¬q) → ¬p is a tautology.
1
Rewrite implications using p → q ≡ ¬p ∨ q
▶
(p → q) ∧ (p → ¬q) → ¬p
≡ [(¬p ∨ q) ∧ (¬p ∨ ¬q)] → ¬p
Inside brackets: (¬p ∨ q) ∧ (¬p ∨ ¬q) = ¬p ∨ (q ∧ ¬q) [distributive]
= ¬p ∨ F [q ∧ ¬q is contradiction = F]
= ¬p [¬p ∨ F ≡ ¬p]
≡ [(¬p ∨ q) ∧ (¬p ∨ ¬q)] → ¬p
Inside brackets: (¬p ∨ q) ∧ (¬p ∨ ¬q) = ¬p ∨ (q ∧ ¬q) [distributive]
= ¬p ∨ F [q ∧ ¬q is contradiction = F]
= ¬p [¬p ∨ F ≡ ¬p]
2
The statement reduces to ¬p → ¬p
▶
The whole statement simplifies to: ¬p → ¬p.
An implication where both sides are the same is always T (p → p is a tautology).
Therefore, the original statement is a tautology. ✅
Intuitive meaning: If p implies both q AND not-q, then p must be false (proof by contradiction).
An implication where both sides are the same is always T (p → p is a tautology).
Therefore, the original statement is a tautology. ✅
Intuitive meaning: If p implies both q AND not-q, then p must be false (proof by contradiction).
Example 2: Checking Argument Validity
Check the validity: "If I go to the gym (G), I feel healthy (H). If I eat well (E), I feel healthy. I don't feel healthy. Therefore, I didn't go to the gym AND I didn't eat well."
1
Translate to symbols
▶
Premises:
1. G → H
2. E → H
3. ¬H
Conclusion: ¬G ∧ ¬E.
From ¬H and G → H: By modus tollens, ¬G is valid.
From ¬H and E → H: By modus tollens, ¬E is valid.
Since both ¬G and ¬E are true, their conjunction ¬G ∧ ¬E is also true.
The argument is VALID. ✅
But wait — in JEE, you need to be careful. The question might ask if the conclusion follows from the premises. Since both modus tollens steps are valid, the conclusion does follow. So the argument is valid.
1. G → H
2. E → H
3. ¬H
Conclusion: ¬G ∧ ¬E.
From ¬H and G → H: By modus tollens, ¬G is valid.
From ¬H and E → H: By modus tollens, ¬E is valid.
Since both ¬G and ¬E are true, their conjunction ¬G ∧ ¬E is also true.
The argument is VALID. ✅
But wait — in JEE, you need to be careful. The question might ask if the conclusion follows from the premises. Since both modus tollens steps are valid, the conclusion does follow. So the argument is valid.
Example 3: Negating Quantified Statements
Write the negation of: "∀x ∈ ℝ, (x² > 0) ∨ (x² = 0)".
1
Apply the negation rule
▶
¬(∀x ∈ ℝ, (x² > 0) ∨ (x² = 0))
≡ ∃x ∈ ℝ, ¬[(x² > 0) ∨ (x² = 0)]
≡ ∃x ∈ ℝ, ¬(x² > 0) ∧ ¬(x² = 0) [De Morgan's]
≡ ∃x ∈ ℝ, (x² ≤ 0) ∧ (x² ≠ 0)
≡ ∃x ∈ ℝ, (x² < 0)
But wait — x² is never negative for real x! So the negation is: "There exists a real x such that x² < 0." This is FALSE. So the original statement is TRUE (which makes sense: every real number squared is either positive or zero).
≡ ∃x ∈ ℝ, ¬[(x² > 0) ∨ (x² = 0)]
≡ ∃x ∈ ℝ, ¬(x² > 0) ∧ ¬(x² = 0) [De Morgan's]
≡ ∃x ∈ ℝ, (x² ≤ 0) ∧ (x² ≠ 0)
≡ ∃x ∈ ℝ, (x² < 0)
But wait — x² is never negative for real x! So the negation is: "There exists a real x such that x² < 0." This is FALSE. So the original statement is TRUE (which makes sense: every real number squared is either positive or zero).
Practice Zone: Think Logically
These questions test if you truly understand mathematical reasoning.