Free Practice Quiz Question List

04 Proof Techniques Online Quiz Questions

Use this free practice quiz with 20 questions to review 04 Proof Techniques, test your knowledge, and prepare for your next test or exam.

20 questions

Choose one

1 point

Which description best characterizes a direct proof of an implication $P\implies Q$ ?

A
Assume the conclusion is false and derive an impossibility.
B
Assume the hypotheses and derive the conclusion.
C
Test several numerical examples and use the pattern as the proof.
D
Prove the converse of the implication.

True or false

1 point

The implication $P\implies Q$ and its contrapositive $\neg Q\implies\neg P$ are logically equivalent.

A
True
B
False

Written response

1 point

What proof method assumes that the statement to be proved is false and then derives an impossibility?

Fill in the blank

1 point

Complete the definition: An integer is even precisely when it can be written as for some integer $k$ .

Choose one

1 point

Which condition must a counterexample satisfy in order to disprove an implication $P\implies Q$ ?

A
An object for which both the hypothesis and conclusion are true.
B
An object for which both the hypothesis and conclusion are false.
C
An object for which the hypothesis is true and the conclusion is false.
D
Any object that makes the conclusion difficult to evaluate.

True or false

1 point

Checking that the first several integers satisfy a universal claim is not, by itself, a proof that the claim holds for every integer.

A
True
B
False

Choose all

1 point

Select all practices recommended for clear mathematical proof writing.

A
State the claim and the domain of the variables clearly.
B
Use a particular numerical example whenever the claim is universal.
C
Define introduced variables such as the integer in $n=2k$ .
D
Explain important transitions instead of giving an unexplained chain of equations.

Written response

1 point

Enter the smallest positive integer counterexample to the claim: If an integer is divisible by $4$ , then it is divisible by $8$ .

Fill in the blank

1 point

Complete the contrapositive of $P\implies Q$ : $\implies$ .

Choose one

1 point

What is the appropriate opening move when proving a statement by contradiction?

A
Assume the conclusion and use it to derive the hypotheses.
B
Assume the statement is false and derive a contradiction.
C
Assume only one convenient example and verify the conclusion.
D
Assume the converse and prove the original implication.

Choose all

1 point

Select all requirements or valid features of a proof by cases.

A
The cases are exhaustive.
B
The conclusion is proved separately in every case.
C
The cases must always be mutually exclusive.
D
The proof explains why the cases cover all relevant possibilities.

Open ended

1 point

Give a proof by contradiction that $\sqrt{2}$ is irrational. Make the initial assumption, justify the parity deductions, state the contradiction, and conclude the claim.

Choose one

1 point

Which statement is the converse of the implication $P\implies Q$ ?

A
If $P$ is false, then $Q$ is false.
B
If $Q$ is false, then $P$ is false.
C
If $Q$ is true, then $P$ is true.
D
If $P$ is true, then $Q$ is false.

True or false

1 point

The proof that $n^2-n$ is even for every integer can validly split into the two cases that $n$ is even and $n$ is odd.

A
True
B
False

Choose one

1 point

Which statement best distinguishes a mathematical proof from a collection of examples?

A
A proof checks only a few representative examples.
B
A proof gives a plausible pattern without justification.
C
A proof is a logically valid argument establishing the conclusion from definitions, assumptions, and established results.
D
A proof is any calculation that produces the expected answer.

Choose one

1 point

For the implication $P\implies Q$ , which statement is its contrapositive?

A
$\neg P\implies\neg Q$
B
$\neg Q\implies\neg P$
C
$Q\implies P$
D
$P\implies\neg Q$

Choose one

1 point

Which opening strategy correctly describes a direct proof of $P\implies Q$ ?

A
Assume $P$ and derive $Q$ .
B
Assume $\neg Q$ and derive $\neg P$ .
C
Assume the conclusion is false and derive a contradiction.
D
Test several numerical examples of $P$ .

Choose one

1 point

In the proof that $\sqrt{2}$ is irrational, what contradiction results from assuming $\sqrt{2}=\frac{p}{q}$ in lowest terms?

A
The equation $p^2=2q^2$ has no integer solutions at all.
B
The assumption implies that $p$ and $q$ are both odd.
C
The assumption directly contradicts the definition of a square root.
D
The assumption implies that both $p$ and $q$ are even, contradicting $\gcd(p,q)=1$ .

Written response

1 point

In the even case of the proof that $n^2-n$ is even, if $n=2k$ , what is the coefficient in the expression $n^2-n=\text{coefficient}\cdot k(n-1)$ ?

Written response

1 point

What mathematical symbol may be placed after the final logical step to indicate that a proof is complete?