Lesson test: Sets

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Sets

1.[1p]
${1, 2, 2} = {2, 1}$ .

True

Correct

False

The answer is: True

2.[3p]
Let $A = {1, {2}}$ . Which statements are true?

Select all that apply

$1 \in A$

Correct

${1} \subseteq A$

Correct

$2 \in A$

The answer is: $1 \in A$ , ${1} \subseteq A$ , ${2} \in A$ , $\emptyset \subseteq A$

${2} \in A$

Correct

${2} \subseteq A$

The answer is: $1 \in A$ , ${1} \subseteq A$ , ${2} \in A$ , $\emptyset \subseteq A$

$\emptyset \subseteq A$

Correct

$\emptyset \in A$

The answer is: $1 \in A$ , ${1} \subseteq A$ , ${2} \in A$ , $\emptyset \subseteq A$

3.[2p]
Why is $\emptyset \subseteq A$ true for every set $A$ ?

For every $x$ the hypothesis of " $x \in \emptyset \Rightarrow x \in A$ " is false, so the conditional is vacuously true

Correct

Every set contains $\emptyset$ as one of its elements

The answer is: For every $x$ the hypothesis of " $x \in \emptyset \Rightarrow x \in A$ " is false, so the conditional is vacuously true

$\emptyset$ is defined to be a subset of every set, with no proof possible

The answer is: For every $x$ the hypothesis of " $x \in \emptyset \Rightarrow x \in A$ " is false, so the conditional is vacuously true

Some element of $\emptyset$ lies in $A$ , which is all a subset needs

The answer is: For every $x$ the hypothesis of " $x \in \emptyset \Rightarrow x \in A$ " is false, so the conditional is vacuously true

4.[2p]
Match each set to the condition for $x$ to belong to it, where complements are taken in a universe $U$ .

$A \cup B$
$A \cap B$
$A ∖ B$
$(A \cup B)^{c}$

$x \in U$ , $x \notin A$ and $x \notin B$
$x \in A$ or $x \in B$
$x \in A$ and $x \notin B$
$x \in A$ and $x \in B$

Show the answer

$A \cup B$ : $x \in A$ or $x \in B$ $A \cap B$ : $x \in A$ and $x \in B$ $A ∖ B$ : $x \in A$ and $x \notin B$ $(A \cup B)^{c}$ : $x \in U$ , $x \notin A$ and $x \notin B$

5.[2p]
In the universe $U = {1, 2, 3, 4, 5, 6}$ , let $A = {1, 2, 3, 4}$ and $B = {3, 4, 5}$ . What is $A^{c} \cup B^{c}$ ?

${1, 2, 5, 6}$

Correct

${6}$

The answer is: ${1, 2, 5, 6}$

${3, 4}$

The answer is: ${1, 2, 5, 6}$

${1, 2, 3, 4, 5}$

The answer is: ${1, 2, 5, 6}$

6.[2p]
Put the lines of the proof that $(A \cup B)^{c} \subseteq A^{c} \cap B^{c}$ in order.

Then $x \in U$ and $x \notin A \cup B$ .
If $x$ were in $A$ or in $B$ , it would be in $A \cup B$ , so $x \notin A$ and $x \notin B$ .
Hence $x \in A^{c}$ and $x \in B^{c}$ .
Let $x \in (A \cup B)^{c}$ .
So $x \in A^{c} \cap B^{c}$ , and since $x$ was arbitrary, the inclusion holds.

Show the answer

a, b, c, d, e

7.[1p]
A set $A$ has $6$ elements. How many elements does $P (A)$ have?

CorrectNot quite: 64

8.[2p]
A set $A$ has $3$ elements. How many elements does $P (A) \times A$ have?

CorrectNot quite: 24

9.[2p]
Applied to $R_{A} = {x \in A : x \notin x}$ for a set $A$ , Russell's argument proves what?

$R_{A} \notin A$ , so no set contains every set

Correct

$R_{A}$ is empty for every set $A$

The answer is: $R_{A} \notin A$ , so no set contains every set

Set builder notation is contradictory even with a surrounding set

The answer is: $R_{A} \notin A$ , so no set contains every set

$R_{A} \in R_{A}$

The answer is: $R_{A} \notin A$ , so no set contains every set

10.[1p]
Let $A = {1, 2, 3, 4, 5}$ and $B = {4, 5, 6, 7}$ . How many elements does the symmetric difference $A △ B$ have?

CorrectNot quite: 5

11.[3p]
For sets $A$ and $B$ , which of these conditions are equivalent to $A \subseteq B$ ?

Select all that apply

$A \cup B = B$

Correct

$A \cap B = A$

Correct

$A ∖ B = \emptyset$

Correct

$B ∖ A = \emptyset$

The answer is: $A \cup B = B$ , $A \cap B = A$ , $A ∖ B = \emptyset$ , $P (A) \subseteq P (B)$

$P (A) \subseteq P (B)$

Correct

12.[2p]
What is $⋂_{n \in ℕ} (0, \frac{1}{n})$ ?

$\emptyset$

Correct

${0}$

The answer is: $\emptyset$

$(0, 1)$

The answer is: $\emptyset$

$[0, 1]$

The answer is: $\emptyset$

13.[2p]
Which axiom of Zermelo and Fraenkel set theory prevents Russell's set ${x : x \notin x}$ from being formed?

Separation: a property may only carve a subset out of a set already given

Correct

Extensionality: sets with the same elements are equal

The answer is: Separation: a property may only carve a subset out of a set already given

Pairing: for any $a$ and $b$ there is a set ${a, b}$

The answer is: Separation: a property may only carve a subset out of a set already given

Power set: every set has a set of all its subsets

The answer is: Separation: a property may only carve a subset out of a set already given

1.[1p] {1,2,2}={2,1}.

2.[3p] Let A={1,{2}}. Which statements are true?

3.[2p] Why is ∅⊆A true for every set A?

4.[2p] Match each set to the condition for x to belong to it, where complements are taken in a universe U.

5.[2p] In the universe U={1,2,3,4,5,6}, let A={1,2,3,4} and B={3,4,5}. What is Ac∪Bc?

6.[2p] Put the lines of the proof that (A∪B)c⊆Ac∩Bc in order.

7.[1p] A set A has 6 elements. How many elements does P(A) have?

8.[2p] A set A has 3 elements. How many elements does P(A)×A have?

9.[2p] Applied to RA={x∈A:x∉x} for a set A, Russell's argument proves what?

10.[1p] Let A={1,2,3,4,5} and B={4,5,6,7}. How many elements does the symmetric difference A△B have?

11.[3p] For sets A and B, which of these conditions are equivalent to A⊆B?

12.[2p] What is ⋂n∈ℕ(0,1n)?

13.[2p] Which axiom of Zermelo and Fraenkel set theory prevents Russell's set {x:x∉x} from being formed?

1.[1p]
${1, 2, 2} = {2, 1}$ .

2.[3p]
Let $A = {1, {2}}$ . Which statements are true?

3.[2p]
Why is $\emptyset \subseteq A$ true for every set $A$ ?

4.[2p]
Match each set to the condition for $x$ to belong to it, where complements are taken in a universe $U$ .

5.[2p]
In the universe $U = {1, 2, 3, 4, 5, 6}$ , let $A = {1, 2, 3, 4}$ and $B = {3, 4, 5}$ . What is $A^{c} \cup B^{c}$ ?

6.[2p]
Put the lines of the proof that $(A \cup B)^{c} \subseteq A^{c} \cap B^{c}$ in order.

7.[1p]
A set $A$ has $6$ elements. How many elements does $P (A)$ have?

8.[2p]
A set $A$ has $3$ elements. How many elements does $P (A) \times A$ have?

9.[2p]
Applied to $R_{A} = {x \in A : x \notin x}$ for a set $A$ , Russell's argument proves what?

10.[1p]
Let $A = {1, 2, 3, 4, 5}$ and $B = {4, 5, 6, 7}$ . How many elements does the symmetric difference $A △ B$ have?

11.[3p]
For sets $A$ and $B$ , which of these conditions are equivalent to $A \subseteq B$ ?

12.[2p]
What is $⋂_{n \in ℕ} (0, \frac{1}{n})$ ?

13.[2p]
Which axiom of Zermelo and Fraenkel set theory prevents Russell's set ${x : x \notin x}$ from being formed?