Contrapositive in Calculus and Number Theory

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

Explanation

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.

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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

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

Explanation

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.

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

Explanation

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.

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

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.

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

Explanation

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

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

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.”

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

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.

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

Explanation

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.”

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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.'

Explanation

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

Explanation

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

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

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.”

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

Explanation

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

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

Explanation

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. 
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14) The negation of "x > 5 and y > 5" is:

Explanation

Negating “A and B” gives “not A or not B.” Using inequalities, “x > 5” negates to “x ≤ 5,” and similarly for y.

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15) Student: "I proved ¬Q → ¬P, so P → Q is true." This reasoning is:

Explanation

The contrapositive of P → Q is ¬Q → ¬P. Showing the contrapositive proves the original statement.

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Which statement is the contrapositive of "If a function f is...
Given "Assume a + b ≤ 10", to complete a contrapositive proof we...
To prove "If x is irrational, then √x is irrational" by...
Proof step: "If a is even or b is even, then ab is even." This...
For "If n ≥ 10, then n² ≥ 100", the contrapositive assumes:
In proving "If prime p > 2, then p is odd" by contrapositive, you'd...
After assuming ¬Q in a contrapositive proof, the next step is to:
Statement: "If f is differentiable, then f is continuous."...
Which error appears in this proof attempt?...
To prove "If x > 0, then x² > 0" by contrapositive, what is the...
The negation of "a and b are both prime" is:
A proof shows "¬P → ¬Q" and concludes "Q → P". This conclusion...
In a proof, “x + y ≠ 10" was assumed in a proof by contrapositive....
The negation of "x > 5 and y > 5" is:
Student: "I proved ¬Q → ¬P, so P → Q is true." This reasoning...
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