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JEE Main Mathematics★ 0 questions in JEE Main from 2024 onward: the NTA removed this unit in its 2024 syllabus revision, alongside Mathematical Induction. Retained here for older question banks, for other entrance papers that still set it, and because the contrapositive is a working proof technique used throughout Calculus

Mathematical Reasoning

An implication is a one-way street. Of its three relatives only the contrapositive says the same thing, and the one everybody reaches for first says something entirely different.

⏱ 19 min readEasy difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

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JEE Main PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Distinguish a statement from a command, a question, an opinion or an open sentence, and negate it correctly
2Form and negate compound statements with and and or, applying De Morgan's laws as the logical twin of the set versions
3Read an implication as not-p or q, build its truth table, and negate it as p and not-q rather than as another implication
4Write the converse, inverse and contrapositive of a statement and identify which are equivalent to the original
5Construct truth tables and classify a compound statement as a tautology, contradiction or contingency
6Negate quantified statements and choose between direct, contrapositive, contradiction and counterexample as a method of proof

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Why this chapter matters in JEE Main

If it rains, the match is cancelled. The match was cancelled, so did it rain? Almost everyone says yes, and the statement never promised that rain is the only thing that can cancel a match. It did not rain, so was the match played? Again no information, since nothing was promised about dry days. But this one is airtight: the match went ahead, therefore it did not rain. Four statements can be built from one implication, and exactly one of the three relatives says the same thing as the original. It is the contrapositive, the one nobody reaches for first. An implication is a one-way street: reversing it gives a different claim, negating it gives a different claim, and only doing both at once returns you to where you started. The NTA removed this unit from JEE Main in 2024, but confusing an implication with its converse remains the most common error in mathematical writing, and proof by contrapositive is a technique you will use whether or not it is examined.

Before you start — revise these

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De Morgan's laws for sets, from Sets, Relations and Functions

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The independent-binary-choice count, which gives a truth table its 2 to the n rows

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Elementary number facts: even and odd, prime, divisibility

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Familiarity with what a proof is expected to look like

Mathematical Reasoning

"If it rains, the match is cancelled."

The match was cancelled. Did it rain?

Almost everyone says yes. But the statement never promised that rain is the only reason a match can be called off. A power failure, a pitch inspection, a strike: any of them cancels the match without a drop falling.

Now the other direction. It did not rain. Was the match played?

Again the answer is no information. Nothing was promised about what happens when it does not rain.

But this one is airtight: the match went ahead, therefore it did not rain. If it had rained, the match would have been cancelled, and it was not.

Statement	Name	Says the same thing?
if $p$ then $q$	the original	—
if $q$ then $p$	converse	no
if not $p$ then not $q$	inverse	no
if not $q$ then not $p$	contrapositive	yes

Four statements, and exactly one of the three relatives is equivalent to the original. It is the contrapositive — the one nobody reaches for first.

An implication is a one-way street. Reversing it gives a different claim, negating it gives a different claim, and only doing both at once gets you back to where you started.

Status. The NTA removed Mathematical Reasoning from the JEE Main syllabus in its 2024 revision, alongside Mathematical Induction. It is kept here because it still appears in older question banks and in several other entrance papers, and because it is short: an hour, not a week. If you are preparing only for JEE Main, read it once for the logic and spend your practice elsewhere.

1. Statements

Statement. A declarative sentence that is either true or false, but not both.

Is a statement	Is not
" $7$ is prime"	"Close the door" (a command)
" $2 + 2 = 5$ " (false, but decidable)	"How old are you?" (a question)
"Every square is a rectangle"	"Mathematics is beautiful" (an opinion)
	" $x + 3 = 7$ " (depends on $x$ )

A false sentence is still a statement. What disqualifies a sentence is being undecidable, not being wrong.

The last row matters: an open sentence containing a free variable becomes a statement only once the variable is fixed or quantified.

2. Negation

The negation of $p$ , written $\sim p$ , is true exactly when $p$ is false.

$\sim (\sim p) \equiv p$

Trap. The negation of "all doors are open" is not "all doors are closed". It is "at least one door is not open", which is a far weaker claim and is what a single counterexample establishes.

3. Compound Statements: And, Or

Conjunction $p \land q$ is true only when both parts are true.

Disjunction $p \lor q$ is true when at least one part is true, including when both are. This is the inclusive or, and it is always the mathematical meaning unless a question says otherwise.

$\sim (p \land q) \equiv \sim p \lor \sim q, \sim (p \lor q) \equiv \sim p \land \sim q$

Those are De Morgan's laws, and they are the same pair met for sets in the first chapter, with intersection replaced by and and union by or. Negating swaps the two connectives.

Illustration 1

Negate: "Ram is tall and Shyam is fair", and "the number is divisible by $2$ or by $3$ ".

Apply De Morgan and watch the connective flip.

$\sim (p \land q) \equiv \sim p \lor \sim q$

"Ram is not tall or Shyam is not fair."

For the second, the negation of an or is an and.

$\sim (p \lor q) \equiv \sim p \land \sim q$

"The number is not divisible by $2$ and not divisible by $3$ ."

Check the second against arithmetic. The original is true for $6$ , $9$ and $10$ ; the negation should be true exactly for numbers like $5$ , $7$ and $11$ , and it is. Had the negation kept the or, it would have been true for $9$ as well, since $9$ is not divisible by $2$ , and both statements would then be true at once, which negation forbids.

4. Implication

$p \Rightarrow q$ : "if $p$ then $q$ ", or equivalently " $p$ only if $q$ ".

$p \Rightarrow q \equiv \sim p \lor q$

That equivalence is the cleanest way to remember the truth table. An implication makes a promise only in the case where $p$ holds, so it is automatically true whenever $p$ is false.

$p$	$q$	$p \Rightarrow q$
T	T	T
T	F	F
F	T	T
F	F	T

Only one row is false, the one where the promise was made and broken.

The nested-circle picture is the whole of it. If $p$ sits inside $q$ , then being outside $q$ guarantees being outside $p$ — which is the contrapositive — while being inside $q$ says nothing about $p$ , which is why the converse fails.

Illustration 2

Negate "if it rains, then the match is cancelled".

The reflex is to negate both parts and keep the if, giving "if it does not rain, the match is not cancelled". That is the inverse, a different statement, not the negation.

Use the equivalence $p \Rightarrow q \equiv \sim p \lor q$ and apply De Morgan.

$\sim (p \Rightarrow q) \equiv \sim (\sim p \lor q) \equiv p \land \sim q$

"It rained and the match was not cancelled."

$the negation of an implication is not an implication at all$

That is exactly right in meaning: the only way to break a promise of the form "if it rains then the match is cancelled" is for it to rain and the match to go ahead. One row of the truth table, and one situation in the world.

Illustration 3

Verify that $p \Rightarrow q$ and $\sim p \lor q$ have the same truth table.

Build both columns side by side; if every row matches, the two are equivalent.

$p$	$q$	$p \Rightarrow q$	$\sim p$	$\sim p \lor q$
T	T	T	F	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

The third and fifth columns agree in all four rows, so the two statements are equivalent.

This is worth more than a verification. It explains why an implication with a false antecedent is true: when $p$ is false, $\sim p$ is true, so the disjunction is true regardless of $q$ . The apparently strange rule that "if $2 + 2 = 5$ then the moon is cheese" counts as a true statement is not a convention at all; it falls straight out of the equivalence.

5. Converse, Inverse, Contrapositive

Name	Form	Equivalent to
Original	$p \Rightarrow q$	contrapositive
Converse	$q \Rightarrow p$	inverse
Inverse	$\sim p \Rightarrow\sim q$	converse
Contrapositive	$\sim q \Rightarrow\sim p$	original

The converse and inverse are equivalent to each other — being each other's contrapositive — and neither is equivalent to the original.

Illustration 4

For "if a number is divisible by $4$ , then it is even", write all three relatives and decide which are true.

Original: divisible by $4$ $\Rightarrow$ even. True.

Converse: even $\Rightarrow$ divisible by $4$ . False, and $6$ proves it.

Inverse: not divisible by $4$ $\Rightarrow$ not even. False, and $6$ proves it again, which is the point: converse and inverse are equivalent, so a single counterexample kills both.

Contrapositive: not even $\Rightarrow$ not divisible by $4$ . True, and necessarily so, since it says the same thing as the original.

$true original + false converse = a one-way implication$

When both an implication and its converse are true, the two statements are equivalent and the connective becomes "if and only if".

Illustration 5

Build the truth table for the biconditional $p \Leftrightarrow q$ , and confirm it is $(p \Rightarrow q) \land (q \Rightarrow p)$ .

The biconditional claims the two statements stand or fall together.

$p$	$q$	$p \Rightarrow q$	$q \Rightarrow p$	both	$p \Leftrightarrow q$
T	T	T	T	T	T
T	F	F	T	F	F
F	T	T	F	F	F
F	F	T	T	T	T

The last two columns agree, so "if and only if" really is the conjunction of an implication and its converse.

Notice the fourth row: both statements false makes the biconditional true, because they still agree. That surprises people, and it is the same feature as an implication being true when its antecedent is false.

In mathematical writing, "if and only if" is a promise that both directions have been checked. A definition is a biconditional; a theorem stating only one direction is not, and reading a one-way theorem as though it were a definition is how false converses enter proofs.

6. Truth Tables, Tautology and Contradiction

A truth table lists every combination of truth values. With $n$ simple statements there are $2^{n}$ rows, which is the independent-binary-choice count from the very first chapter.

Term	Meaning
Tautology	true in every row
Contradiction	false in every row
Contingency	neither

Illustration 6

Show that $(p \land q) \Rightarrow p$ is a tautology, and that $p \land \sim p$ is a contradiction.

Build the table for the first, one column at a time.

$p$	$q$	$p \land q$	$(p \land q) \Rightarrow p$
T	T	T	T
T	F	F	T
F	T	F	T
F	F	F	T

True in all four rows, so it is a tautology. Read in words it is obvious: if both hold then certainly the first holds. Rows three and four are true for the other reason, that the antecedent is false and no promise was made.

For the second, $p$ and $\sim p$ can never be true together, so the conjunction is false in both rows. A contradiction.

$a tautology and its negation is a contradiction, and vice versa$

7. Quantifiers

Symbol	Reading	Negation becomes
$\forall$	for all	there exists ... not
$\exists$	there exists	for all ... not

Negating a quantifier flips it. "All swans are white" is refuted by one black swan, not by every swan being black, which is why the negation is existential.

Illustration 7

Negate: "for every real $x$ , $x^{2} \geq 0$ " and "there exists a prime that is even".

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

"There exists a real $x$ with $x^{2} < 0$ ." This is false, as it should be, since the original is true.

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

"Every prime is odd." This is false too, since $2$ is prime and even, so the original was true.

Note how the burden of proof moves. Proving a for all statement needs an argument covering every case; disproving it needs one example. That asymmetry is why counterexamples are so much cheaper than proofs, and it is entirely a consequence of how negation treats quantifiers.

8. Validating Statements: Methods of Proof

Method	Structure	Use when
Direct	assume $p$ , deduce $q$	the implication is straightforward
Contrapositive	assume $\sim q$ , deduce $\sim p$	the negations are easier to work with
Contradiction	assume $\sim p$ , derive an absurdity	the statement asserts impossibility
Counterexample	one case where it fails	disproving a universal claim

Illustration 8

Prove that if $n^{2}$ is even then $n$ is even.

Directly, you would have to deduce something about $n$ from a fact about $n^{2}$ , which is awkward: knowing $n^{2} = 2 k$ tells you little about $n$ without a square root.

The contrapositive is far easier, and it is a complete proof because the two are equivalent.

$contrapositive: n odd \Rightarrow n^{2} odd$

Let $n = 2 m + 1$ .

$n^{2} = 4 m^{2} + 4 m + 1 = 2 (2 m^{2} + 2 m) + 1$

That is odd, having the form $2 (integer) + 1$ . The contrapositive is proved, so the original is proved.

Choosing the contrapositive is not a weaker argument. It is the same statement, and it was available only because the four relatives were sorted out first.

Illustration 9

Prove that $2$ is irrational.

The claim asserts that something is impossible, which is the signature of a proof by contradiction.

Suppose $2 = p / q$ in lowest terms, so $p$ and $q$ share no common factor.

$2 q^{2} = p^{2} \Rightarrow p^{2} is even \Rightarrow p is even$

That last step is Illustration 6, used as a lemma. Write $p = 2 r$ .

$2 q^{2} = 4 r^{2} \Rightarrow q^{2} = 2 r^{2} \Rightarrow q is even, by the same result$

So $p$ and $q$ are both even, contradicting "lowest terms". The assumption must fail, and $2$ is irrational.

Illustration 10

Decide whether "every odd number greater than $1$ is prime" is true.

A universal claim is disproved by a single case, so look for one before attempting any proof.

$3, 5, 7 prime, 9 = 3 \times 3 not prime$

One counterexample settles it. False.

Notice how little work that took compared with what a proof would have needed. Before trying to prove any "for all" statement, spend thirty seconds looking for a counterexample: if the statement is false, you finish in one line, and if you fail to find one, you have usually learnt why it is true.

9. A Note on Syllabus Status

The NTA removed this unit from the JEE Main Mathematics syllabus in the 2024 revision, together with Mathematical Induction, as part of aligning the paper with the NCERT Class 11 and 12 courses.

Where it still counts	Where it no longer does
older question banks and mock papers	JEE Main, from 2024 onward
several state and other entrance papers
the reasoning behind every proof you will ever write

Keep the chapter for the last row. Nothing here is examined in Main any more, but the distinction between an implication and its converse is the single most common error in mathematical writing, and the contrapositive is a working proof technique you will use in Calculus and Number Theory whether or not it is examined.

Summary

A statement is a declarative sentence that is definitely true or definitely false. Being false does not disqualify it; being undecidable does.

The negation of "all are" is "at least one is not", never "all are not".

De Morgan's laws swap and with or under negation, exactly as they swap intersection with union for sets.

An implication is equivalent to $\sim p \lor q$ , so it is automatically true whenever $p$ is false. Only one row of its truth table is false.

The negation of an implication is not an implication: it is $p \land \sim q$ , the single case of a promise made and broken.

Of the three relatives, only the contrapositive is equivalent to the original. Converse and inverse are equivalent to each other and to neither.

One counterexample therefore kills both the converse and the inverse at once.

An implication is true whenever its antecedent is false, which falls straight out of the equivalence with $\sim p \lor q$ rather than being a convention.

When an implication and its converse are both true, the statement becomes an "if and only if".

A truth table has $2^{n}$ rows. A tautology is true in all of them, a contradiction in none.

Negating a quantifier flips it, which is why disproving a universal claim costs one example while proving it costs an argument.

Proof by contrapositive is a complete proof, not a weaker one, and it is usually chosen because the negations are easier to handle.

Before proving any "for all" statement, spend thirty seconds hunting a counterexample.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

The organising principle

an implication is a one-way street, and only its contrapositive says the same thing

Reversing gives the converse, negating gives the inverse, and neither is equivalent to the original. Doing both at once returns you to where you started.

What counts as a statement

a declarative sentence that is definitely true or definitely false, but not both

Being false does not disqualify a sentence; being undecidable does. An open sentence with a free variable becomes a statement only once the variable is fixed or quantified.

Negation

not-p is true exactly when p is false; not-not-p is p

The negation of all doors are open is at least one door is not open, NOT all doors are closed. The second is a far stronger claim than negation permits.

Conjunction and disjunction

p and q needs both true; p or q needs at least one, including both

The or is always inclusive in mathematics unless stated otherwise. That is why the negation of an or is an and rather than another or.

De Morgan's laws for statements

not(p and q) = not-p or not-q; not(p or q) = not-p and not-q

The same pair met for sets, with intersection reading as and and union as or. Negating swaps the two connectives every time.

Implication as a disjunction

p implies q is equivalent to not-p or q

This is why an implication is automatically true whenever p is false: not-p is then true and the disjunction holds regardless of q. The rule is a consequence, not a convention.

Truth table of an implication

true in three rows, false only when p is true and q is false

The single false row is the one where a promise was made and broken, which is also the only situation the negation describes.

Negation of an implication

not(p implies q) is p and not-q

The negation of an implication is NOT an implication. Negating both halves and keeping the if produces the inverse, which is a different statement entirely.

The four related conditionals

original and contrapositive are equivalent; converse and inverse are equivalent to each other

Diagonals of the square agree, sides do not. One counterexample therefore kills both the converse and the inverse at once.

Biconditional

p if and only if q is (p implies q) and (q implies p)

True when the two statements agree, including when both are false. A definition is a biconditional; a one-way theorem is not, and reading one as the other is how false converses enter proofs.

Truth tables and classification

n simple statements give 2 to the n rows; a tautology is true in all, a contradiction in none

The row count is the independent-binary-choice argument from the first chapter. The negation of a tautology is a contradiction and vice versa.

Negating quantifiers

not(for all x, P) is (there exists x, not-P); not(there exists x, P) is (for all x, not-P)

Negating a quantifier flips it. This is why disproving a universal claim costs one example while proving it costs an argument covering every case.

Methods of proof

direct, contrapositive, contradiction, or a single counterexample to disprove

Contrapositive is a COMPLETE proof, not a weaker one. Contradiction suits claims of impossibility. Before proving any for-all statement, hunt a counterexample for thirty seconds.

JEE Main test series

Ready to be tested on this?

Mathematical Reasoning shows up in every JEE Main paper — sit a timed one and find out where you actually stand.

~1Est. Qs

Mediumpriority

Full JEE Main mocks, real clock
Mathematics-only sectionals & chapter tests
All-India rank + topic-level analysis

Free forever · solver-verified questions

⚠️

Traps JEE Main sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Treating the converse as though it followed from the original

✓ If it rains the match is cancelled says nothing about why an uncancelled match was cancelled. Divisible by 4 implies even is true and its converse is false, with 6 as the witness. Only when both directions are separately established may you write if and only if.

Why it happens: In everyday speech if often carries the sense of if and only if, so the reversal feels implied.

WATCH OUT

✗ Negating an implication into another implication

✓ Use the equivalence with not-p or q and apply De Morgan: the negation is p and not-q. For the rain example it is it rained and the match was not cancelled, which is a single situation rather than a rule. Negating both halves and keeping the if gives the inverse, a different statement.

Why it happens: The statement has an if in it, so its negation looks as though it should too.

WATCH OUT

✗ Negating all are into all are not

✓ Negating a quantifier flips it. The negation of all swans are white is there exists a swan that is not white, which one black swan establishes. All swans are black is much stronger and is not what negation gives.

Why it happens: The word not is attached to the predicate rather than to the whole claim.

WATCH OUT

✗ Treating the exclusive or as the default

✓ In mathematics or is inclusive: p or q is true when both hold. That is what makes the De Morgan pair work, and a question wanting the exclusive sense will say either ... or ... but not both.

Why it happens: Everyday usage of or usually excludes the both case, as in tea or coffee.

WATCH OUT

✗ Calling a proof by contrapositive an indirect or weaker argument

✓ The contrapositive is logically identical to the original, so proving it proves the original outright. For n squared even implies n even, the contrapositive route is three lines while the direct route stalls immediately, and nothing about the conclusion is weaker.

Why it happens: It does not follow the shape of assume p, deduce q, so it feels like a substitute.

WATCH OUT

✗ Attempting to prove a universal statement without first testing it

✓ Spend thirty seconds looking for a counterexample first. Every odd number above 1 is prime survives 3, 5 and 7 and dies at 9. If the statement is false you finish in one line, and if you fail to find a counterexample you usually discover why it is true.

Why it happens: The statement is presented as something to prove, so its truth is assumed.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Mathematical Reasoning?

12 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

12 questions~8 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Removed from the JEE Main syllabus in the 2024 NTA revision, along with Mathematical Induction
•An implication is a one-way street; only the contrapositive is equivalent to it
•Converse and inverse are equivalent to each other, so one counterexample kills both
•A false sentence is still a statement; an undecidable one is not
•The negation of all are is at least one is not, never all are not
•De Morgan swaps and with or, exactly as it swaps intersection with union
•p implies q is not-p or q, which is why a false antecedent makes it true
•The negation of an implication is p and not-q, not another implication
•The or of mathematics is inclusive unless a question says otherwise
•A truth table has 2 to the n rows; tautology true in all, contradiction in none
•Negating a quantifier flips it, which is why counterexamples are cheap and proofs are not
•Contrapositive proof is complete, not weaker; hunt a counterexample before proving anything universal

JEE Main question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Question style	Marks each	Typical count	What it tests
Statements, negation and compound statements	0	0
Implication, converse and contrapositive	0	0
Truth tables, tautology and quantifiers	0	0
Methods of proof and counterexamples	0	0

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Since the unit is off the JEE Main syllabus from 2024, read it once for the logic and do not build a practice schedule around it. If you meet one of these questions in an older paper, the four-conditional square answers most of them in seconds.
For any negation question, write the statement symbolically first. Almost every wrong answer here comes from negating in words, where the connectives are easy to lose.
Draw the square of four conditionals whenever converse or contrapositive appears. The diagonals are equivalent and the sides are not, and that single picture answers the whole family of questions.
Check any proposed negation by finding a case where the original is true. The negation must be false there, and this catches the all-are-not error immediately.
For a universal claim, test small cases before attempting to prove it. Finding a counterexample finishes the question in one line, and failing to find one usually shows you the structure of the proof.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Every legal contract and specification document lives on …

Every legal contract and specification document lives on the distinction this chapter draws: a clause saying if the goods are damaged then a refund is due promises nothing about refunds for undamaged goods, and disputes routinely turn on someone reading the converse

Software testing is counterexample hunting

Software testing is counterexample hunting, since a program is a universal claim that it behaves correctly on every input, and one failing case refutes it while a thousand passing cases prove nothing

Medical and scientific reporting depends on not confusing…

Medical and scientific reporting depends on not confusing an implication with its converse: a symptom that follows from a disease does not mean the disease follows from the symptom, which is the same asymmetry that Bayes' theorem quantifies in the Probability chapter

Where else this topic is tested

Prepare once, score in every exam that asks it.

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Questions aspirants ask

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Read it once, then spend your practice time elsewhere. The NTA removed the unit in the 2024 revision together with Mathematical Induction, so no JEE Main question will test it directly. Three reasons to keep it anyway. Older question banks and mock papers still contain these questions, and meeting one without knowing the vocabulary is unsettling. Several other entrance papers still set the material. And most importantly, the distinction between an implication and its converse is the single most common error in mathematical writing, while proof by contrapositive is a working technique you will use in Calculus, in Number Theory and in any proof-based question. The chapter is short, an hour rather than a week, and it pays for itself in the other units.

Because of the equivalence with not-p or q. When p is false, not-p is true, so the whole disjunction is true regardless of q, and that is the entire reason. The rule is a consequence of the definition rather than a convention imposed on it. The intuition is that an implication is a promise about what happens when p holds; if p never holds, the promise was never tested and cannot have been broken. This is why the statement if 2 plus 2 equals 5 then the moon is cheese counts as true, which feels odd only until you notice that nothing was ever claimed about the moon in any situation that actually occurs. The single false row of the truth table is the only one where a promise was made and broken.

Work outwards, applying one rule at a time. Negating a conjunction gives a disjunction of the negations and vice versa, which is De Morgan. Negating an implication first requires rewriting it as not-p or q, after which De Morgan gives p and not-q. Negating a quantifier flips it, so a for-all becomes a there-exists and the predicate inside is then negated in turn. The most reliable check is truth: a correct negation must be false whenever the original is true, so test the pair on a case where you know the answer. Applying that check to all doors are open immediately rejects all doors are closed, since both are false when exactly one door is shut, and two statements cannot both be false if one is the negation of the other.

Contradiction suits statements that assert impossibility or non-existence, where there is no natural implication to reverse. That there is no largest prime, or that root 2 is irrational, are both claims that something cannot happen, and the only handle available is to assume it does and follow the consequences until something breaks. The contrapositive suits genuine implications where the negated forms are easier to manipulate than the originals, as with n squared even implies n even, where starting from an odd n gives a clean substitution while starting from an even square gives nothing to hold on to. A practical hint: if the statement contains no, never or is irrational, reach for contradiction; if it has a clear if-then shape, try the contrapositive first.

They answer opposite questions, and the asymmetry comes entirely from how negation treats quantifiers. To establish a statement of the form for all x, P holds, you need an argument covering every case, since one omission is fatal. To refute the same statement you need only one x where P fails, because the negation is existential. That is why disproof is so much cheaper than proof, and why testing a universal claim on many cases proves nothing: the expression n squared plus n plus 41 is prime for every n from 1 to 39 and fails at 40. Conversely, to establish a there-exists statement you need one example, while refuting it requires an argument about every case. Before attempting any proof, work out which of these four situations you are in.