Contrapositive in Calculus and Number Theory

Reviewed by Alva Benedict B.

Alva Benedict B., PhD

College Expert

Review Board Member

Alva Benedict B. is an experienced mathematician and math content developer with over 15 years of teaching and tutoring experience across high school, undergraduate, and test prep levels. He specializes in Algebra, Calculus, and Statistics, and holds advanced academic training in Mathematics with extensive expertise in LaTeX-based math content development.

, PhD

By Thames

Thames

Community Contributor

Quizzes Created: 8156 | Total Attempts: 9,588,805

| Questions: 15 | Updated: Jan 28, 2026

Share on Facebook Share on Twitter Share on Whatsapp Share on Pinterest Share on Email Copy to Clipboard Embed on your website

Question 1 / 16

🏆 Rank #-- ▾

0 %

0/100

Score 0/100

1) Which statement is the contrapositive of "If a function f is differentiable at x = c, then f is continuous at x = c"?

If a function f is continuous at x = c, then f is differentiable at x = c

If a function f is not continuous at x = c, then f is not differentiable at x = c

If a function f is not differentiable at x = c, then f is not continuous at x = c

If a function f is discontinuous at x = c, then f is differentiable at x = c

The contrapositive of "If P, then Q" is "If not Q, then not P." For the statement "If a function f is differentiable at x = c, then f is continuous at x = c," P is "f is differentiable at x = c" and Q is "f is continuous at x = c." The negation of Q is "f is not continuous at x = c" and the negation of P is "f is not differentiable at x = c." Therefore, the contrapositive is "If a function f is not continuous at x = c, then f is not differentiable at x = c." This is a fundamental result in calculus. Option A is the converse, option C is the inverse, and option D is logically incorrect.

Explanation

Submit

About This Quiz

Contrapositive In Calculus and Number Theory - Quiz

Think you can handle proofs that mix logic with calculus and algebra? This quiz lets you test that by working with classic results like “differentiable implies continuous,” inequalities involving squares, and number-theoretic statements about primes and products. You’ll decide when it’s easier to assume the negation of the conclusion (like... see morea function being discontinuous or a product failing a property) and then logically work back to the negation of the hypothesis. Step by step, you’ll see how contrapositive proofs help untangle tricky statements that would be hard to prove directly.
see less

2)

What first name or nickname would you like us to use?

You may optionally provide this to label your report, leaderboard, or certificate.

2) Given "Assume a + b ≤ 10", to complete a contrapositive proof we must conclude:

Cannot determine without more context

A > 10 or b > 10

A ≤ 5 or b ≤ 5

A < 5 or b < 5

Without knowing the original statement “If P, then Q,” we cannot know what ¬Q is supposed to look like. The assumption “a + b ≤ 10” could correspond to ¬Q, but the needed conclusion (¬P) depends entirely on the original implication. Therefore, more information is required.

Explanation

Submit

3) To prove "If x is irrational, then √x is irrational" by contrapositive, start with:

Assume x is irrational

Assume √x is rational

Assume √x is irrational

Assume x is rational

The contrapositive of “If x is irrational, then √x is irrational” is: “If √x is rational, then x is rational.” Therefore the proof begins by assuming √x is rational and attempting to show x must also be rational.

Explanation

Submit

4) Proof step: "If a is even or b is even, then ab is even." This supports:

If ab is odd, then a and b are both odd (contrapositive)

If ab is even, then a or b is even (direct)

If a and b are odd, then ab is odd (converse)

If a is odd, then ab is odd

The contrapositive of “If ab is odd, then a and b are both odd” is “If a is even or b is even, then ab is even.” Showing this contrapositive establishes the truth of the original statement.

Explanation

The contrapositive of “If ab is odd, then a and b are both odd” is “If a is even or b is even, then ab is even.” Showing this contrapositive establishes the truth of the original statement.

Submit

5) For "If n ≥ 10, then n² ≥ 100", the contrapositive assumes:

N² < 100

N < 10

N² ≤ 99

Both A and C are correct

The contrapositive negates the conclusion and then implies the negation of the hypothesis. The negation of “n² ≥ 100” is “n²

Explanation

The contrapositive negates the conclusion and then implies the negation of the hypothesis. The negation of “n² ≥ 100” is “n²

Submit

6) In proving "If prime p > 2, then p is odd" by contrapositive, you'd show:

If p is even, then p ≤ 2 or p is not prime

If p is odd, then p > 2

If p is even and prime, then p = 2

If p ≤ 2, then p is not odd

The contrapositive of “If p > 2 and p is prime, then p is odd” is: “If p is even, then p ≤ 2 or p is not prime.” This captures the logical negation of “p > 2 and prime.”

Explanation

The contrapositive of “If p > 2 and p is prime, then p is odd” is: “If p is even, then p ≤ 2 or p is not prime.” This captures the logical negation of “p > 2 and prime.”

Submit

7) After assuming ¬Q in a contrapositive proof, the next step is to:

Show Q is true

Show P is true

Derive ¬P using logical steps

Derive a contradiction

In proving P → Q by contrapositive, we assume ¬Q and work logically to derive ¬P. Showing a contradiction is not part of contrapositive method—that belongs to proof by contradiction.

Explanation

In proving P → Q by contrapositive, we assume ¬Q and work logically to derive ¬P. Showing a contradiction is not part of contrapositive method—that belongs to proof by contradiction.

Submit

8) Statement: "If f is differentiable, then f is continuous." Contrapositive proves this by showing:

Continuous functions are differentiable

Discontinuous functions are not differentiable

Differentiable functions are continuous

Non-differentiable functions are discontinuous

The negation of “f is continuous” is “f is discontinuous.” The contrapositive negates the conclusion and shows the negation of the hypothesis: “If f is not continuous, then f is not differentiable.”

Explanation

Submit

9) Which error appears in this proof attempt? 'To prove "If n is odd, then n² is odd," I assume n² is even and state without proof that n must therefore be even. Therefore, by contrapositive, if n is odd, n² is odd.'

Correct contrapositive proof

Assumes what needs to be proved

Circular reasoning

Proves the converse

The student attempts a contrapositive proof of "If n is odd, then n² is odd" by proving "If n² is even, then n is even." But in concluding "Therefore, if n is odd, n² is odd", they implicitly assume that "n is even" and "n is odd" are the only possibilities (law of excluded middle), which is valid. The error lies in the unstated assumption that the properties of even/odd are mutually exclusive and exhaustive without establishing this. More critically, the proof assumes that "n² is even" implies "n is even" without justification, which is precisely what needs to be proved in the contrapositive. This shows the student assuming the key implication rather than proving it.

Explanation

Submit

10) To prove "If x > 0, then x² > 0" by contrapositive, what is the correct starting assumption?

X ≤ 0

X² ≤ 0

X² < 0 or x² = 0

Both A and C are correct

A contrapositive proof requires assuming the negation of the conclusion. The conclusion is “x² > 0”, so our assumption should be x² ≤ 0.

Explanation

A contrapositive proof requires assuming the negation of the conclusion. The conclusion is “x² > 0”, so our assumption should be x² ≤ 0.

Submit

11) The negation of "a and b are both prime" is:

A and b are both composite

A is composite and b is composite

A is composite or b is composite

Neither a nor b is prime

The negation of “A and B” is “not A or not B.” Thus, “both prime” negates to “at least one is not prime,” which is “a is composite or b is composite.”

Explanation

The negation of “A and B” is “not A or not B.” Thus, “both prime” negates to “at least one is not prime,” which is “a is composite or b is composite.”

Submit

12) A proof shows "¬P → ¬Q" and concludes "Q → P". This conclusion is:

Correct; that's contrapositive equivalence

Wrong; should conclude P → Q

Wrong; should conclude ¬Q → ¬P

Correct; that's the converse

“¬P → ¬Q” is the contrapositive of “Q → P.” So concluding “Q → P” is valid.

Explanation

“¬P → ¬Q” is the contrapositive of “Q → P.” So concluding “Q → P” is valid.

Submit

13) In a proof, “x + y ≠ 10" was assumed in a proof by contrapositive. What's the original conclusion?

X + y > 10

X + y = 10

Cannot determine from assumption alone

X + y < 10

In a contrapositive proof of 'If P, then Q', we assume ¬Q. Here, ¬Q is 'x+y ≠ 10'. The logical negation of 'not equal to' is 'equal to'. Therefore, the original conclusion Q in the implication must have been 'x+y = 10'. Note that this reasoning holds regardless of what P is, as we're only concerned with identifying Q from ¬Q.

Explanation

Submit