Lesson test: Order

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Order

1.[2p]
Which of these relations are partial orders?

Select all that apply

$\leq$ on $ℝ$

Correct

$\subseteq$ on $P (X)$ , for a set $X$

Correct

Divisibility on $ℕ$

Correct

Divisibility on $ℤ$

The answer is: $\leq$ on $ℝ$ , $\subseteq$ on $P (X)$ , for a set $X$ , Divisibility on $ℕ$

$<$ on $ℝ$

The answer is: $\leq$ on $ℝ$ , $\subseteq$ on $P (X)$ , for a set $X$ , Divisibility on $ℕ$

" $x + y$ is even" on $ℤ$

The answer is: $\leq$ on $ℝ$ , $\subseteq$ on $P (X)$ , for a set $X$ , Divisibility on $ℕ$

2.[1p]
A relation that is symmetric cannot also be antisymmetric.

True

The answer is: False

False

Correct

3.[2p]
Divisibility on $ℤ$ is not a partial order. Which property fails, and what shows it?

Antisymmetry: $2 ∣ - 2$ and $- 2 ∣ 2$ , yet $2 \neq - 2$

Correct

Reflexivity: $0$ does not divide $0$

The answer is: Antisymmetry: $2 ∣ - 2$ and $- 2 ∣ 2$ , yet $2 \neq - 2$

Transitivity: $2 ∣ 4$ and $4 ∣ - 8$ , yet $2$ does not divide $- 8$

The answer is: Antisymmetry: $2 ∣ - 2$ and $- 2 ∣ 2$ , yet $2 \neq - 2$

Totality: $2$ and $3$ are incomparable

The answer is: Antisymmetry: $2 ∣ - 2$ and $- 2 ∣ 2$ , yet $2 \neq - 2$

4.[2p]
Which of these are total orders?

Select all that apply

$\leq$ on $ℚ$

Correct

Divisibility on ${1, 2, 4, 8}$

Correct

Divisibility on ${1, 2, 3, 6}$

The answer is: $\leq$ on $ℚ$ , Divisibility on ${1, 2, 4, 8}$

$\subseteq$ on $P ({1, 2})$

The answer is: $\leq$ on $ℚ$ , Divisibility on ${1, 2, 4, 8}$

5.[2p]
The Hasse diagram of the divisors of $30$ under divisibility draws one line for each covering pair. How many lines does it have?

CorrectNot quite: 12

6.[3p]
Match each poset to the description that fits it.

${1, 2, 3, 4, 6}$ under divisibility
The divisors of $12$ under divisibility
$ℕ$ under $\leq$
$ℤ$ under $\leq$

a greatest element, which is the only maximal one
a least element and no maximal element
no maximal and no minimal element
two maximal elements and no greatest element

Show the answer

${1, 2, 3, 4, 6}$ under divisibility: two maximal elements and no greatest element The divisors of $12$ under divisibility: a greatest element, which is the only maximal one $ℕ$ under $\leq$ : a least element and no maximal element $ℤ$ under $\leq$ : no maximal and no minimal element

7.[1p]
What is $\sup {1 - \frac{1}{n} : n \in ℕ}$ in $ℝ$ ?

$1$ , which is not an element of the set

Correct

$1$ , which is the greatest element of the set

The answer is: $1$ , which is not an element of the set

$\frac{1}{2}$ , the largest element that can be written down

The answer is: $1$ , which is not an element of the set

It does not exist, because the set has no greatest element

The answer is: $1$ , which is not an element of the set

8.[3p]
Let $T = {q \in ℚ : q^{2} < 2}$ , ordered by $\leq$ . Which statements are true?

Select all that apply

$T$ is bounded above in $ℚ$

Correct

$1.42$ is an upper bound of $T$

Correct

$1.414$ is an upper bound of $T$

The answer is: $T$ is bounded above in $ℚ$ , $1.42$ is an upper bound of $T$ , Viewed as a subset of $ℝ$ , $T$ has supremum $\sqrt{2}$

$T$ has a least upper bound in $ℚ$

The answer is: $T$ is bounded above in $ℚ$ , $1.42$ is an upper bound of $T$ , Viewed as a subset of $ℝ$ , $T$ has supremum $\sqrt{2}$

Viewed as a subset of $ℝ$ , $T$ has supremum $\sqrt{2}$

Correct

9.[2p]
Put the lines of the proof that $\subseteq$ is transitive in order.

Let $A \subseteq B$ and $B \subseteq C$ .
Since $x$ was arbitrary, $A \subseteq C$ .
Let $x \in A$ .
Since $B \subseteq C$ , $x \in C$ .
Since $A \subseteq B$ , $x \in B$ .

Show the answer

a, b, c, d, e

10.[1p]
In $(ℕ, ∣)$ , what is the meet $12 \land 18$ , the infimum of ${12, 18}$ ?

$6$ , the greatest common divisor

Correct

$36$ , the least common multiple

The answer is: $6$ , the greatest common divisor

$12$ , the smaller of the two

The answer is: $6$ , the greatest common divisor

It does not exist, because $12$ and $18$ are incomparable

The answer is: $6$ , the greatest common divisor

11.[3p]
Which of these are well-orders?

Select all that apply

$\leq$ on $ℕ$

Correct

$\leq$ on $ℤ$

The answer is: $\leq$ on $ℕ$ , The lexicographic order on $ℕ \times ℕ$ , The shortlex order on finite words in the letters $a$ and $b$

$\leq$ on ${q \in ℚ : q \geq 0}$

The answer is: $\leq$ on $ℕ$ , The lexicographic order on $ℕ \times ℕ$ , The shortlex order on finite words in the letters $a$ and $b$

The lexicographic order on $ℕ \times ℕ$

Correct

The dictionary order on finite words in the letters $a$ and $b$

The answer is: $\leq$ on $ℕ$ , The lexicographic order on $ℕ \times ℕ$ , The shortlex order on finite words in the letters $a$ and $b$

The shortlex order on finite words in the letters $a$ and $b$

Correct

12.[2p]
What is the largest number of elements in an antichain of ${1, 2, \dots, 12}$ under divisibility?

CorrectNot quite: 6

13.[1p]
Every finite non-empty poset has a minimal element.

True

Correct

False

The answer is: True

14.[2p]
Zorn's lemma says that a non-empty poset in which every non-empty chain has an upper bound has what?

A maximal element

Correct

A greatest element

The answer is: A maximal element

A least element

The answer is: A maximal element

A linear extension that is a well-order

The answer is: A maximal element

1.[2p] Which of these relations are partial orders?

2.[1p] A relation that is symmetric cannot also be antisymmetric.

3.[2p] Divisibility on ℤ is not a partial order. Which property fails, and what shows it?

4.[2p] Which of these are total orders?

5.[2p] The Hasse diagram of the divisors of 30 under divisibility draws one line for each covering pair. How many lines does it have?

6.[3p] Match each poset to the description that fits it.

7.[1p] What is sup{1−1n:n∈ℕ} in ℝ?

8.[3p] Let T={q∈ℚ:q2<2}, ordered by ≤. Which statements are true?

9.[2p] Put the lines of the proof that ⊆ is transitive in order.

10.[1p] In (ℕ,∣), what is the meet 12∧18, the infimum of {12,18}?

11.[3p] Which of these are well-orders?

12.[2p] What is the largest number of elements in an antichain of {1,2,…,12} under divisibility?

13.[1p] Every finite non-empty poset has a minimal element.

14.[2p] Zorn's lemma says that a non-empty poset in which every non-empty chain has an upper bound has what?

1.[2p]
Which of these relations are partial orders?

2.[1p]
A relation that is symmetric cannot also be antisymmetric.

3.[2p]
Divisibility on $ℤ$ is not a partial order. Which property fails, and what shows it?

4.[2p]
Which of these are total orders?

5.[2p]
The Hasse diagram of the divisors of $30$ under divisibility draws one line for each covering pair. How many lines does it have?

6.[3p]
Match each poset to the description that fits it.

7.[1p]
What is $\sup {1 - \frac{1}{n} : n \in ℕ}$ in $ℝ$ ?

8.[3p]
Let $T = {q \in ℚ : q^{2} < 2}$ , ordered by $\leq$ . Which statements are true?

9.[2p]
Put the lines of the proof that $\subseteq$ is transitive in order.

10.[1p]
In $(ℕ, ∣)$ , what is the meet $12 \land 18$ , the infimum of ${12, 18}$ ?

11.[3p]
Which of these are well-orders?

12.[2p]
What is the largest number of elements in an antichain of ${1, 2, \dots, 12}$ under divisibility?

13.[1p]
Every finite non-empty poset has a minimal element.

14.[2p]
Zorn's lemma says that a non-empty poset in which every non-empty chain has an upper bound has what?