Lesson test: Direct proof and proof by cases

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Direct proof and proof by cases

1.[1p]
How should a direct proof of "for every integer $n$ , if $n$ is odd then $n^{2}$ is odd" begin?

Let $n$ be an odd integer.

Correct

Let $n = 3$ , which is odd.

The answer is: Let $n$ be an odd integer.

Suppose $n^{2}$ is odd.

The answer is: Let $n$ be an odd integer.

Let $n$ be an integer, and suppose $n$ is not odd.

The answer is: Let $n$ be an odd integer.

2.[2p]
Which of these statements are true?

Select all that apply

$3 ∣ 12$

Correct

$12 ∣ 3$

The answer is: $3 ∣ 12$ , $- 4 ∣ 12$ , $7 ∣ 0$ , $1 ∣ 13$

$- 4 ∣ 12$

Correct

$7 ∣ 0$

Correct

$0 ∣ 7$

The answer is: $3 ∣ 12$ , $- 4 ∣ 12$ , $7 ∣ 0$ , $1 ∣ 13$

$1 ∣ 13$

Correct

3.[2p]
Put the lines of this proof that if $a ∣ b$ and $b ∣ c$ then $a ∣ c$ in order.

By definition, $b = a k$ and $c = b m$ for some integers $k$ and $m$ .
Let $a$ , $b$ and $c$ be integers with $a ∣ b$ and $b ∣ c$ .
Substituting, $c = (a k) m = a (k m)$ .
Since $k$ and $m$ are integers, so is $k m$ .
So $c$ is $a$ times an integer, and by definition $a ∣ c$ .

Show the answer

a, b, c, d, e

4.[1p]
A proof of $| x + y | \leq | x | + | y |$ that treats only the case $x \geq 0, y \geq 0$ and the case $x < 0, y < 0$ is complete.

True

The answer is: False

False

Correct

5.[2p]
The proof that every odd square is one more than a multiple of $8$ writes $n = 2 k + 1$ and gets $n^{2} = 4 (k^{2} + k) + 1$ . What finishes it?

The earlier result that $k^{2} + k$ is even for every integer $k$

Correct

The fact that $k$ must itself be odd

The answer is: The earlier result that $k^{2} + k$ is even for every integer $k$

Checking that $1$ , $9$ , $25$ and $49$ are each one more than a multiple of $8$

The answer is: The earlier result that $k^{2} + k$ is even for every integer $k$

The fact that $4$ divides $8$

The answer is: The earlier result that $k^{2} + k$ is even for every integer $k$

6.[3p]
Match each claim to the method its proof in the lesson used.

If $a ∣ b$ and $b ∣ c$ then $a ∣ c$
$n^{2} + n$ is even for every integer $n$
$| x + y | \leq | x | + | y |$ for all reals
For every $k$ there are $k$ consecutive non-primes
Some irrational $a$ and $b$ have $a^{b}$ rational

cases on signs
a non-constructive existence proof
a constructive existence proof
a direct proof from the definition
cases on parity

Show the answer

If $a ∣ b$ and $b ∣ c$ then $a ∣ c$ : a direct proof from the definition $n^{2} + n$ is even for every integer $n$ : cases on parity $| x + y | \leq | x | + | y |$ for all reals: cases on signs For every $k$ there are $k$ consecutive non-primes: a constructive existence proof Some irrational $a$ and $b$ have $a^{b}$ rational: a non-constructive existence proof

7.[3p]
The proof that some irrational $a$ and $b$ give $a^{b}$ rational considers ${\sqrt{2}}^{\sqrt{2}}$ . Which of these are true of it?

Select all that apply

It shows that one of two pairs works without saying which

Correct

It uses the fact that every real number is rational or irrational

Correct

It shows that ${\sqrt{2}}^{\sqrt{2}}$ is rational

The answer is: It shows that one of two pairs works without saying which, It uses the fact that every real number is rational or irrational, It relies on $\sqrt{2}$ being irrational

It names a pair that works

The answer is: It shows that one of two pairs works without saying which, It uses the fact that every real number is rational or irrational, It relies on $\sqrt{2}$ being irrational

It relies on $\sqrt{2}$ being irrational

Correct

8.[2p]
The constructive proof takes $N = (k + 1)!$ and the numbers $N + 2$ to $N + (k + 1)$ . For $k = 5$ , which run of non-primes does it produce?

$722$ to $726$

Correct

$24$ to $28$

The answer is: $722$ to $726$

$122$ to $126$

The answer is: $722$ to $726$

$720$ to $724$

The answer is: $722$ to $726$

9.[2p]
Why does a direct proof of "if $n^{2}$ is even then $n$ is even" stall?

The hypothesis $n^{2} = 2 k$ gives $n = \pm \sqrt{2 k}$ , which is not visibly twice an integer

Correct

The statement is false for negative $n$

The answer is: The hypothesis $n^{2} = 2 k$ gives $n = \pm \sqrt{2 k}$ , which is not visibly twice an integer

Its hypothesis is false for every integer $n$

The answer is: The hypothesis $n^{2} = 2 k$ gives $n = \pm \sqrt{2 k}$ , which is not visibly twice an integer

A direct proof cannot be used on a statement about every integer

The answer is: The hypothesis $n^{2} = 2 k$ gives $n = \pm \sqrt{2 k}$ , which is not visibly twice an integer

10.[1p]
What is the remainder given by the division algorithm when $- 23$ is divided by $7$ ?

CorrectNot quite: 5

11.[2p]
In the uniqueness proof of the division algorithm, why does $0 \leq b (q - q') < b$ , with $b > 0$ , force $q = q'$ ?

The only integer $m$ with $0 \leq b m < b$ is $m = 0$

Correct

By the well-ordering principle

The answer is: The only integer $m$ with $0 \leq b m < b$ is $m = 0$

Because $r$ and $r'$ are both non-negative

The answer is: The only integer $m$ with $0 \leq b m < b$ is $m = 0$

Because $b$ divides $a$

The answer is: The only integer $m$ with $0 \leq b m < b$ is $m = 0$

12.[2p]
A student proves $x^{2} + y^{2} \geq 2 x y$ by writing: " $x^{2} + y^{2} \geq 2 x y$ , so $x^{2} - 2 x y + y^{2} \geq 0$ , so $(x - y)^{2} \geq 0$ , which is true." What is wrong?

It reasons from the claim to a true statement, which proves nothing about the claim

Correct

The claim is false when $x$ and $y$ are negative

The answer is: It reasons from the claim to a true statement, which proves nothing about the claim

$(x - y)^{2} \geq 0$ is not true for every pair of reals

The answer is: It reasons from the claim to a true statement, which proves nothing about the claim

Nothing: the proof is complete as written

The answer is: It reasons from the claim to a true statement, which proves nothing about the claim

13.[3p]
Which of these pairs of proofs together establish $P ⟺ Q$ ?

Select all that apply

A proof of $P \Rightarrow Q$ and a proof of $Q \Rightarrow P$

Correct

A proof of $P \Rightarrow Q$ and a proof of $\neg P \Rightarrow \neg Q$

Correct

A proof of $Q \Rightarrow P$ and a proof of $\neg Q \Rightarrow \neg P$

Correct

A proof of $P \Rightarrow Q$ and a proof of $\neg Q \Rightarrow \neg P$

The answer is: A proof of $P \Rightarrow Q$ and a proof of $Q \Rightarrow P$ , A proof of $P \Rightarrow Q$ and a proof of $\neg P \Rightarrow \neg Q$ , A proof of $Q \Rightarrow P$ and a proof of $\neg Q \Rightarrow \neg P$

Two different proofs of $P \Rightarrow Q$

14.[1p]
"For all real numbers $x$ and $y$ , $y - x \geq 0$ " can be proved by saying "without loss of generality $x \leq y$ ".

True

The answer is: False

False

Correct

1.[1p] How should a direct proof of "for every integer n, if n is odd then n2 is odd" begin?

2.[2p] Which of these statements are true?

3.[2p] Put the lines of this proof that if a∣b and b∣c then a∣c in order.

4.[1p] A proof of |x+y|≤|x|+|y| that treats only the case x≥0,y≥0 and the case x<0,y<0 is complete.

5.[2p] The proof that every odd square is one more than a multiple of 8 writes n=2k+1 and gets n2=4(k2+k)+1. What finishes it?

6.[3p] Match each claim to the method its proof in the lesson used.

7.[3p] The proof that some irrational a and b give ab rational considers 22. Which of these are true of it?

8.[2p] The constructive proof takes N=(k+1)! and the numbers N+2 to N+(k+1). For k=5, which run of non-primes does it produce?

9.[2p] Why does a direct proof of "if n2 is even then n is even" stall?

10.[1p] What is the remainder given by the division algorithm when −23 is divided by 7?

11.[2p] In the uniqueness proof of the division algorithm, why does 0≤b(q−q′)<b, with b>0, force q=q′?

12.[2p] A student proves x2+y2≥2xy by writing: "x2+y2≥2xy, so x2−2xy+y2≥0, so (x−y)2≥0, which is true." What is wrong?

13.[3p] Which of these pairs of proofs together establish P⟺Q?

14.[1p] "For all real numbers x and y, y−x≥0" can be proved by saying "without loss of generality x≤y".

1.[1p]
How should a direct proof of "for every integer $n$ , if $n$ is odd then $n^{2}$ is odd" begin?

2.[2p]
Which of these statements are true?

3.[2p]
Put the lines of this proof that if $a ∣ b$ and $b ∣ c$ then $a ∣ c$ in order.

4.[1p]
A proof of $| x + y | \leq | x | + | y |$ that treats only the case $x \geq 0, y \geq 0$ and the case $x < 0, y < 0$ is complete.

5.[2p]
The proof that every odd square is one more than a multiple of $8$ writes $n = 2 k + 1$ and gets $n^{2} = 4 (k^{2} + k) + 1$ . What finishes it?

6.[3p]
Match each claim to the method its proof in the lesson used.

7.[3p]
The proof that some irrational $a$ and $b$ give $a^{b}$ rational considers ${\sqrt{2}}^{\sqrt{2}}$ . Which of these are true of it?

8.[2p]
The constructive proof takes $N = (k + 1)!$ and the numbers $N + 2$ to $N + (k + 1)$ . For $k = 5$ , which run of non-primes does it produce?

9.[2p]
Why does a direct proof of "if $n^{2}$ is even then $n$ is even" stall?

10.[1p]
What is the remainder given by the division algorithm when $- 23$ is divided by $7$ ?

11.[2p]
In the uniqueness proof of the division algorithm, why does $0 \leq b (q - q') < b$ , with $b > 0$ , force $q = q'$ ?

12.[2p]
A student proves $x^{2} + y^{2} \geq 2 x y$ by writing: " $x^{2} + y^{2} \geq 2 x y$ , so $x^{2} - 2 x y + y^{2} \geq 0$ , so $(x - y)^{2} \geq 0$ , which is true." What is wrong?

13.[3p]
Which of these pairs of proofs together establish $P ⟺ Q$ ?

14.[1p]
"For all real numbers $x$ and $y$ , $y - x \geq 0$ " can be proved by saying "without loss of generality $x \leq y$ ".