Direct Proof Basics: From Hypothesis to Conclusion

Reviewed by Editorial Team

The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.

Learn about Our Editorial Process

| By Thames

Thames

Community Contributor

Quizzes Created: 7387 | Total Attempts: 9,527,684

| Questions: 15 | Updated: Dec 1, 2025

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

Question 1 / 15

0 %

0/100

Score 0/100

1) What is the first step in a direct proof of "If P, then Q"?

Assume Q is true

Assume P is true

Assume P is false

Assume Q is false

In a direct proof of a conditional statement "If P, then Q", the goal is to show that P implies Q. The first step is to assume that P is true. Then, using logical deductions, definitions, and known facts, we derive that Q must be true. This approach directly establishes the implication.

Explanation

Submit

About This Quiz

Direct Proof Basics: From Hypothesis To Conclusion - Quiz

Are you ready to see how formal proofs actually work step by step? In this quiz, you’ll practice the core idea of a direct proof: start by assuming the hypothesis is true and then use logic, definitions, and simple algebra to reach the conclusion. You’ll work with even and odd... see morenumbers, divisibility, and basic inequalities to see how each statement flows into the next. By the end, you’ll feel more confident reading and writing direct proofs without feeling lost in symbols! 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) In a direct proof of "If n is even, then n² is even," what do we assume first?

N² is even

N is odd

N² is odd

N is even

For a direct proof, we start by assuming the hypothesis. Here, the hypothesis is that n is even. So, we assume n is even, meaning n = 2k for some integer k. Then, we compute n² = (2k)² = 4k² = 2(2k²), which is even, thus deriving the conclusion.

Explanation

Submit

3) Which statement best describes a direct proof?

Starting from premises and logically deriving the conclusion

Proving a statement by showing its contrapositive is true

Assuming the negation and finding a contradiction

Breaking the proof into separate cases

A direct proof involves starting with the given hypotheses or premises and then using logical steps, definitions, and theorems to directly arrive at the conclusion. This method does not involve assuming the negation or using contrapositive, but rather builds a chain of reasoning from the assumptions to the conclusion.

Explanation

Submit

4) In a direct proof of "If x > 0, then x² > 0," we start by assuming:

X² > 0

X² ≤ 0

X > 0

X ≤ 0

In a direct proof, we assume the hypothesis. Here, the hypothesis is x > 0. Since x is positive, multiplying both sides by x (which is positive) gives x² > 0, thus directly proving the statement.

Explanation

In a direct proof, we assume the hypothesis. Here, the hypothesis is x > 0. Since x is positive, multiplying both sides by x (which is positive) gives x² > 0, thus directly proving the statement.

Submit

5) Which of these is NOT a characteristic of a direct proof?

Uses contradiction

Uses logical implications

Starts with the hypothesis

Ends with the conclusion

A direct proof relies on logical implications from the hypothesis to the conclusion and does not use contradiction. Using contradiction is a feature of proof by contradiction, where we assume the negation of the conclusion and derive a false statement.

Explanation

Submit

6) To prove "If a number is divisible by 4, then it is even" directly, we start by:

Assuming the number is even

Assuming the number is divisible by 4

Assuming the number is not divisible by 4

Assuming the number is odd

In a direct proof, we assume the hypothesis. Here, the hypothesis is that the number is divisible by 4. From this, we can show that since 4 is a multiple of 2, the number is also divisible by 2, meaning it is even.

Explanation

Submit

7) In a direct proof of "If n is odd, then n+1 is even," the first line should be:

"Assume n+1 is even."

"Suppose n is not odd."

"Assume n is odd."

"Let n+1 be odd."

The first step in a direct proof is to assume the hypothesis. Here, the hypothesis is that n is odd. Therefore, we begin with "Assume n is odd." Then, since n is odd, n+1 is even, which can be shown directly.

Explanation

Submit

8) Which proof technique directly shows P → Q by assuming P and deriving Q?

Contradiction

Contrapositive

Counterexample

Direct proof

Direct proof is the technique where we assume P is true and then through logical steps derive that Q is true. This directly establishes the implication P → Q. Other techniques like contradiction or contrapositive do not involve assuming P and deriving Q directly.

Explanation

Submit

9) To directly prove "If x and y are even, then x+y is even," we begin by:

Assuming x and y are even

Assuming x+y is even

Assuming x or y is odd

Assuming x+y is odd

In a direct proof, we start with the hypothesis that x and y are even. This means x = 2a and y = 2b for some integers a and b. Then, x+y = 2a + 2b = 2(a+b), which is even, thus proving the statement.

Explanation

In a direct proof, we start with the hypothesis that x and y are even. This means x = 2a and y = 2b for some integers a and b. Then, x+y = 2a + 2b = 2(a+b), which is even, thus proving the statement.

Submit

10) What is the correct format for the first line of a direct proof of "If A, then B"?

"Assume B is true."

"Assume A is true."

"Suppose A is false."

"Suppose B is false."

The standard first line in a direct proof of "If A, then B" is to assume that A is true. This sets up the hypothesis, and then we proceed to show that B must be true under this assumption.

Explanation

The standard first line in a direct proof of "If A, then B" is to assume that A is true. This sets up the hypothesis, and then we proceed to show that B must be true under this assumption.

Submit

11) In a direct proof that "If n is divisible by 6, then n is divisible by 3," what do we assume first?

N is divisible by 3

N is not divisible by 6

N is divisible by 6

N is not divisible by 3

For a direct proof, we assume the hypothesis, which is that n is divisible by 6. Since 6 is divisible by 3, it follows that n is also divisible by 3, thus proving the conclusion.

Explanation

For a direct proof, we assume the hypothesis, which is that n is divisible by 6. Since 6 is divisible by 3, it follows that n is also divisible by 3, thus proving the conclusion.

Submit

12) In a direct proof of "If a = b and b = c, then a = c," we start by:

Assuming a = c

Assuming a ≠ b or b ≠ c

Assuming a ≠ c

Assuming a = b and b = c

This is a direct proof of the transitive property of equality. We start by assuming the hypothesis, which is that a = b and b = c. Then, by substitution, we can conclude that a = c.

Explanation

This is a direct proof of the transitive property of equality. We start by assuming the hypothesis, which is that a = b and b = c. Then, by substitution, we can conclude that a = c.

Submit

13) Which of these statements is best suited for a direct proof?

"If n is even, then n² is even"

"There is no largest prime number"

"√2 is irrational"

"Not all continuous functions are differentiable"

Direct proofs are well-suited for conditional statements of the form "If P, then Q" where we can assume P and derive Q through logical steps. The statement "If n is even, then n² is even" is such a conditional statement and can be proven directly by assuming n is even and showing n² is even.

Explanation

Submit

14) To directly prove "If x is rational and y is rational, then x+y is rational," we begin by:

Assuming x+y is rational

Assuming x and y are rational

Assuming x or y is irrational

Assuming x+y is irrational

In a direct proof, we assume the hypothesis that x and y are rational. We use the definition of rational numbers, this means x = a/b and y = c/d for some integers a,b,c,d with b,d ≠ 0. Then, x+y = (ad + bc)/(bd), which is rational, thus proving the statement.

Explanation

Submit

15) In the direct proof that "If n is a multiple of 4, then n² is a multiple of 16," what is the first assumption?

N² = 16m for some integer m

N is not a multiple of 4

N = 4k for some integer k

N² is not a multiple of 16

We start by assuming the hypothesis that n is a multiple of 4, which means n = 4k for some integer k. Then, n² = (4k)² = 16k², which is a multiple of 16, thus proving the conclusion.

Explanation

We start by assuming the hypothesis that n is a multiple of 4, which means n = 4k for some integer k. Then, n² = (4k)² = 16k², which is a multiple of 16, thus proving the conclusion.

Submit