Lesson test: What a proof is

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What a proof is

1.[2p]
What does a proof supply that checking many cases does not?

A reason the claim cannot fail in any case at all

Correct

Confidence that no counterexample is likely

The answer is: A reason the claim cannot fail in any case at all

A method for computing the answer faster

The answer is: A reason the claim cannot fail in any case at all

Evidence that the claim holds for large numbers

The answer is: A reason the claim cannot fail in any case at all

2.[2p]
How many counterexamples are needed to refute a claim about every natural number?

One

Correct

Infinitely many

The answer is: One

As many as were checked in its favour

The answer is: One

None, since such claims cannot be refuted

The answer is: One

3.[3p]
$2^{p} - 1$ is prime for $p = 2, 3, 5, 7$ . At which prime $p$ does it first fail?

CorrectNot quite: 11

4.[2p]
The contrapositive of "if $P$ then $Q$ " is

if not $Q$ then not $P$

Correct

if $Q$ then $P$

The answer is: if not $Q$ then not $P$

if not $P$ then not $Q$

The answer is: if not $Q$ then not $P$

$P$ and $Q$ are equivalent

The answer is: if not $Q$ then not $P$

5.[3p]
In the proof that $\sqrt{2}$ is irrational, where is the contradiction reached?

$p$ and $q$ were taken with no common factor, and both turn out to be even

Correct

$p^{2} = 2 q^{2}$ has no integer solutions

The answer is: $p$ and $q$ were taken with no common factor, and both turn out to be even

The square root of a fraction is never a fraction

The answer is: $p$ and $q$ were taken with no common factor, and both turn out to be even

$2$ is prime, so it has no square root

The answer is: $p$ and $q$ were taken with no common factor, and both turn out to be even

6.[2p]
Euclid's proof of the infinitude of primes shows that $p_{1} p_{2} \dots p_{k} + 1$ is always prime.

True

The answer is: False

False

Correct

7.[3p]
Put the parts of a proof by induction in order.

Prove the base case, that $S (1)$ holds
State the claim $S (n)$ to be proved for every $n$
Assume $S (n)$ for a particular $n$
Deduce $S (n + 1)$ from that assumption

Show the answer

a, b, c, d

8.[3p]
The claim $2^{n} > n^{2}$ needs a base case above $n = 1$ . At which value of $n$ does the induction start?

CorrectNot quite: 5

9.[3p]
Which of these are honest limits on what a proof gives you?

Select all that apply

It establishes the conclusion only from its assumptions, which may not describe anything real

Correct

A long proof can contain an error and later be repaired

Correct

No system strong enough for arithmetic proves every true statement about the naturals

Correct

A proved theorem may stop being true once better measurements are made

The answer is: It establishes the conclusion only from its assumptions, which may not describe anything real, A long proof can contain an error and later be repaired, No system strong enough for arithmetic proves every true statement about the naturals

1.[2p] What does a proof supply that checking many cases does not?

2.[2p] How many counterexamples are needed to refute a claim about every natural number?

3.[3p] 2p−1 is prime for p=2,3,5,7. At which prime p does it first fail?

4.[2p] The contrapositive of "if P then Q" is

5.[3p] In the proof that 2 is irrational, where is the contradiction reached?

6.[2p] Euclid's proof of the infinitude of primes shows that p1p2⋯pk+1 is always prime.

7.[3p] Put the parts of a proof by induction in order.

8.[3p] The claim 2n>n2 needs a base case above n=1. At which value of n does the induction start?

9.[3p] Which of these are honest limits on what a proof gives you?

1.[2p]
What does a proof supply that checking many cases does not?

2.[2p]
How many counterexamples are needed to refute a claim about every natural number?

3.[3p]
$2^{p} - 1$ is prime for $p = 2, 3, 5, 7$ . At which prime $p$ does it first fail?

4.[2p]
The contrapositive of "if $P$ then $Q$ " is

5.[3p]
In the proof that $\sqrt{2}$ is irrational, where is the contradiction reached?

6.[2p]
Euclid's proof of the infinitude of primes shows that $p_{1} p_{2} \dots p_{k} + 1$ is always prime.

7.[3p]
Put the parts of a proof by induction in order.

8.[3p]
The claim $2^{n} > n^{2}$ needs a base case above $n = 1$ . At which value of $n$ does the induction start?

9.[3p]
Which of these are honest limits on what a proof gives you?