Proof by Contradiction Quiz: Expose Logical Inconsistencies

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| Questions: 20 | Updated: Dec 17, 2025

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1) What is the defining feature of a proof by contradiction?

You assume the negation and derive an impossibility

You avoid assumptions entirely

You prove the statement directly

You rely only on examples

A contradiction proof begins by assuming the opposite of the target conclusion. Logical deductions then force an impossible outcome—showing the assumed negation cannot be true and the original statement must hold.

Explanation

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About This Quiz

Proof By Contradiction Quiz: Expose Logical Inconsistencies - Quiz

Proof by contradiction can feel like a clever puzzle, and this contradictions proof quiz lets you practice the method in a clear, intuitive way. You’ll assume the opposite of what you want to prove, follow the consequences, and watch how the logic collapses into an impossible situation. Along the way,... see moreyou’ll learn why contradictions are so effective in uncovering truth and how they help simplify complicated statements.
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2) To prove 'If ab is even, then at least one of a or b is even' by contradiction, what should be assumed?

Ab is even and both a and b are odd

Ab is even and both a and b are even

Ab is odd and both a and b are odd

Ab is odd and both a and b are even

We assume the hypothesis holds (ab is even) while the conclusion fails (both a and b are odd). Since odd·odd = odd, this leads to an immediate contradiction.

Explanation

We assume the hypothesis holds (ab is even) while the conclusion fails (both a and b are odd). Since odd·odd = odd, this leads to an immediate contradiction.

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3) In the classical proof that √2 is irrational, where does the contradiction arise?

A/b is not in lowest terms

√2 is actually an integer

A and b are not integers

√2 is not a real number

Assuming √2 = a/b in lowest terms forces both a and b to be even after algebraic manipulation. This contradicts the assumption that the fraction was simplified.

Explanation

Assuming √2 = a/b in lowest terms forces both a and b to be even after algebraic manipulation. This contradicts the assumption that the fraction was simplified.

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4) Why is proof by contradiction useful?

It can succeed when direct proof is difficult

It avoids all logical reasoning

It only applies to false statements

It proves statements without assumptions

Some statements are hard to prove directly, but assuming their negation creates a structure that quickly leads to an impossibility. This indirect route can simplify the argument.

Explanation

Some statements are hard to prove directly, but assuming their negation creates a structure that quickly leads to an impossibility. This indirect route can simplify the argument.

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5) To prove 'There is no smallest positive rational number' by contradiction, what do we assume?

There exists a smallest positive rational number

All rationals are positive

Some rationals are negative

Rationals form a finite set

Begin by assuming such a smallest rational exists. Constructing a smaller positive rational (like r/2) breaks the assumption, giving a contradiction.

Explanation

Begin by assuming such a smallest rational exists. Constructing a smaller positive rational (like r/2) breaks the assumption, giving a contradiction.

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6) In a proof by contradiction, what should the assumption be?

The negation of the statement being proved

A true statement

A random related claim

The original statement itself

Contradiction requires assuming the opposite of the goal. If this assumption leads to an impossibility, the original statement must be true.

Explanation

Contradiction requires assuming the opposite of the goal. If this assumption leads to an impossibility, the original statement must be true.

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7) What role does logic play in proof by contradiction?

Deriving consequences until an impossibility appears

Avoiding all structured reasoning

Proving the assumption is true

Listing examples

Logical rules transform the assumed negation into further consequences. When these consequences contradict known facts, the assumption collapses.

Explanation

Logical rules transform the assumed negation into further consequences. When these consequences contradict known facts, the assumption collapses.

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8) Which classical statement is proven by contradiction?

√2 is irrational

The sum of angles in a triangle is 180°

The Pythagorean theorem

Prime numbers are finite

The irrationality of √2 is famously established by contradiction: assuming rationality forces a violation of lowest-term representation.

Explanation

The irrationality of √2 is famously established by contradiction: assuming rationality forces a violation of lowest-term representation.

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9) After finding a contradiction in a proof, what must be concluded?

The assumption was false

The assumption was true

The contradiction should be ignored

The proof must restart

Once a contradiction arises from the assumed negation, the assumption cannot be correct. Therefore the original statement must be true.

Explanation

Once a contradiction arises from the assumed negation, the assumption cannot be correct. Therefore the original statement must be true.

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10) To prove 'The product of two consecutive integers is even' by contradiction, what is assumed?

The product is odd

The integers are not consecutive

The product is even

Both integers are odd

Assuming the product is odd forces both integers to be odd. But two odd integers differ by 2, not 1, contradicting that they are consecutive.

Explanation

Assuming the product is odd forces both integers to be odd. But two odd integers differ by 2, not 1, contradicting that they are consecutive.

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11) In Euclid’s proof of infinitely many primes, what contradiction appears?

There are no primes

The constructed number is divisible by all primes

The constructed number is itself prime

The constructed number has a prime factor not in the assumed list

Assuming only finitely many primes exist, Euclid constructs a new number whose prime factorization cannot use solely the listed primes, contradicting finiteness.

Explanation

Assuming only finitely many primes exist, Euclid constructs a new number whose prime factorization cannot use solely the listed primes, contradicting finiteness.

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12) What is an advantage of proof by contradiction?

It analyzes what follows if the conclusion fails, often exposing impossibilities

It avoids logic entirely

It always yields the shortest proof

It guarantees intuitive understanding

By assuming the negation, the proof often uncovers structural constraints that quickly become impossible, aiding in proving the original claim.

Explanation

By assuming the negation, the proof often uncovers structural constraints that quickly become impossible, aiding in proving the original claim.

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13) Which of the following is a valid contradiction?

0 = 0

1 = 1

2 = 3

X = x

A contradiction must violate a fundamental truth. The equation 2 = 3 cannot occur under standard arithmetic, making it a valid contradiction indicator.

Explanation

A contradiction must violate a fundamental truth. The equation 2 = 3 cannot occur under standard arithmetic, making it a valid contradiction indicator.

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14) In proof by contradiction, what must be true of the assumption?

It must be the negation of the statement being proved

It must be obviously true

It must be impossible

It must be unrelated

Assuming the direct negation of the statement gives a clear way to show it leads to inconsistency, proving the original statement.

Explanation

Assuming the direct negation of the statement gives a clear way to show it leads to inconsistency, proving the original statement.

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15) The usual proof that √2 is irrational is an example of which method?

Proof by contradiction

Direct proof

Induction

Proof by construction

The argument assumes rationality, manipulates the expression, and reaches an impossibility in parity, confirming contradiction.

Explanation

The argument assumes rationality, manipulates the expression, and reaches an impossibility in parity, confirming contradiction.

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16) Which type of statement is commonly proven using contradiction?

Uniqueness or impossibility statements

Basic arithmetic identities

Geometric constructions

Simple algebraic equalities

Contradiction is especially effective when showing that something cannot exist or only one possibility is viable. The method works by temporarily assuming the opposite of what you want to prove and demonstrating that this assumption leads to an impossibility. When the assumption collapses logically, the original statement becomes the only viable truth, making contradiction perfect for uniqueness proofs or statements ruling out certain outcomes.

Explanation

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17) In contradiction proofs about parity, which contradiction is often used?

A number is both even and odd

1 = 1

Even numbers always end with 5

Odd numbers are divisible by 2

Showing a number is simultaneously even and odd violates essential parity rules, providing a clear contradiction. Even numbers are defined as multiples of 2, while odd numbers differ by exactly 1 from an even number. No integer can satisfy both definitions simultaneously, so assuming both properties at once yields an immediate logical collapse that ends the argument.

Explanation

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18) To prove 'If 3n + 2 is odd, then n is odd' by contradiction, what do we assume?

3n + 2 is odd and n is even

3n + 2 is even and n is odd

3n + 2 is odd and n is odd

3n + 2 is even and n is even

We assume the hypothesis is true but the conclusion false. Letting n be even means n = 2k, so 3n+2 becomes 6k+2, clearly divisible by 2 and therefore even. This contradicts the assumption that 3n+2 is odd, proving the original implication must hold because the attempt to negate it leads to inconsistency.

Explanation

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19) In proving irrationality of √3 by contradiction, what key step occurs?

Showing both numerator and denominator must be divisible by 3

Showing √3 is an integer

Showing √3 equals 3

Showing rational numbers are finite

By squaring both sides and rearranging, the equation leads to 3 dividing a and, subsequently, 3 dividing b. But a/b was assumed to be in simplest form—having both numerator and denominator divisible by 3 contradicts that assumption, proving √3 is irrational.

Explanation

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20) Why is proof by contradiction considered logically valid?

It avoids logical reasoning

It only works for inequalities

It depends on constructing examples

It relies on the law of excluded middle: either a statement or its negation must be true

If assuming ¬P leads to an impossibility, then ¬P cannot hold. Thus P must be true, consistent with classical logic principles. This relies on the law of noncontradiction: a statement and its negation cannot both be true. If the negation yields a contradiction, it must be false, and the original statement P becomes logically forced as the only remaining consistent option.

Explanation

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