COMP 256 Midterm Studyguide

[!links] Parent: COMP 256 MOCCOMP 256 MOC\[!links\] \_Home MOC References LaTeX Reference Class Notes cn256_IntegersAndDivision_2026-03-24 Midterm COMP 256 Midterm Studyguide

Info

Thursday, March 26th
Held on canvas

Topics

Sets
Sequences (sums/loops)
Proofs
Counting

Previous Quiz Questions

Quiz 1

Question 1

$$\text{If } |A| = 4, \text {what is } |P(P(A))|$$

$2^{16}$
$16$
$32$
$2^4$

[!Answer]- $2^{16}$

Explanation: The number of elements in a power set is always $2^{|S|}$. $|P(A)| = 2^{4} \implies |P(P(A))| = 2^{2^4} = 2^{16}$

Question 2

$$\text{Which statements are always true for set} A? \text{ (select all correct)}$$

$\emptyset \subseteq a$
$\emptyset \subset A$
$A \subseteq \emptyset$
$A \subseteq A$
$A \subset A$

[!Answer]-

$\emptyset \subseteq a$

$A \subseteq A$

Explanation:

By definition, the empty set $\emptyset$ is a subset of every set.

Every set is a subset of itself

Question 3

$$\text {Which sets have exactly 2 elements? (select all correct)}$$

${{1,2}}$
${ \emptyset , { \emptyset }}$
${{\emptyset}}$
${1,2}$
${{1},{2}}$

[!Answer]-

${ \emptyset , { \emptyset }}$

${1,2}$

${{1},{2}}$

Explanation: Look at the curly-braces and figure it out

Question 4

$$\text {Which of the following is true? (select all correct)}$$

$\emptyset \times A = A$
$A \times B = B \times A$
$A \times \emptyset = \emptyset$
$|A \times B| = |B \times A|$

[!Answer]-

$A \times \emptyset = \emptyset$

You are essentially multiplying A with zero, so you get an empty set.

$|A \times B| = |B \times A|$

Since you are comparing cardinality (size) the order of the elements in the pairs doesn't matter; therefore they are equal.

Question 5

$$\text {Which statements are always true? (select all correct)}$$

$|P(A)| = |A|$
$|P(A)| = 2^{|A|}$
$|P(A)| = |A|^2$
$|P(A)| = 2|A|$

[!Answer]-

$|P(A)| = 2^{|A|}$

Asking for the formula for the cardinality of a power set, formula used in Question 1COMP 256 Midterm Studyguide\[!links\] Parent: COMP 256 MOC Info * Thursday, March 26th * Held on canvas Topics * Sets * Sequences (sums/loops) * Proofs * Counting Previous Quiz Questions Quiz 1 Question 1 $$\text{If } |A| = 4, \text {what is } |P(P(A))|$$ * $2^{16}$ * $16$ * $32$ * $2^4$ \[!Answer\]- $2^{16}$ Explanation: The number of elements in a power set is always $2^{|S|}$. $|P(A)| = 2^{4} \implies |P(P(A))| = 2^{2^4} = 2^{16}$ Question 2 $$\text{Which statements are always true for set} A? \text{ (sel

Question 6

$$\text{If } |A| = |B| \text{, what is true? (Select all correct answers)}$$

$A \cap B \neq \emptyset$
$|A \cup B| = |A|$
$\text{none}$
$|A \cap B| = |A|$
$A = B$

[!Answer]- $\text{none}$

Question 7

$$\text{If } A = \emptyset \text{, what is true? (Select all that are correct)}$$

$\emptyset \in A$
$\emptyset \subseteq A$
$|A| = 1$
$A \subset \emptyset$

[!Answer]-

$\emptyset \subseteq A$

Question 8

$$\text{Which of the following are elements of } P({1, 2})\text{? (Select all that are correct)}$$

$\emptyset$
${1, 2}$
$1$
${2}$
${1}$

[!Answer]-

$\emptyset$

${1, 2}$

${2}$

${1}$

Question 9

$$\text{If } A={a, b, c}, \text{ how many elements does } P(A) \text{ have?}$$

[!Answer]-

$8$

Question 10

$$\text{Which are always subsets of } A \cup B \text{? (Select all correct answers)}$$

$B - A$
$A - B$
$A \cap B$
$A$
$B$

[!Answer]-

$B - A$

$A - B$

$A \cap B$

$A$

$B$

Question 11

$$\text{Which expressions describe the same region? (Select all correct answers)}$$

$A - (A \cap B)$
$A \cap B$
$A - B$
$A \cap B^C$

[!Answer]-

$A - (A \cap B)$

$A - B$

$A \cap B^C$

Question 12

$$\text{If } A \subset B, \text{ which must be true? (Select all that are correct)}$$

$A \subseteq B$
$A \neq B$
$|A| < |B|$
$B \subseteq A$

[!Answer]-

$A \subseteq B$

$A \neq B$

$|A| < |B|$

Question 13

$$\text{Let } A={1, 2, 3, 4} \text{ and } B={3, 4, 5}. \text{ What is } A - B?$$

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

[!Answer]-

${1, 2}$

Question 14

$$\text{Which are true for all sets?}$$

$\emptyset - A = \emptyset$
$A \cap \emptyset = \emptyset$
$A \cup \emptyset = A$
$A - \emptyset = \emptyset$

[!Answer]-

$\emptyset - A = \emptyset$

$A \cap \emptyset = \emptyset$

$A \cup \emptyset = A$

Question 15

$$\text{Which situation is impossible? (Select all that are correct)}$$

$|A| = 8, |B| = 8, |A \cap B| = 0$
$|A| = 5, |B| = 7, |A \cap B| = 3$
$|A| = 6, |B| = 6, |A \cap B| = 6$
$|A| = 10, |B| = 12, |A \cap B| = 15$

[!Answer]-

$|A| = 10, |B| = 12, |A \cap B| = 15$

Question 16

$$\text{Evaluate } |P(C)| \cdot |A \times B| = ?$$

$32$

[!Answer]-

$32$

Question 17

$$\text{Let } A = {{1, 2}, 3}. \text{ Which is true? (Select all that are correct)}$$

$2 \in A$
${3} \in A$
$1 \in A$
${1, 2} \in A$

[!Answer]-

${1, 2} \in A$

Question 18

$$\text{Which implications are always true? (Select all that are correct)}$$

$A \subseteq B \implies A \cup B = B$
$A \cap B = A \implies A \subseteq B$
$A \subseteq B \implies A \cap B = A$
$A \cup B = A \implies B \subseteq A$

[!Answer]-

$A \subseteq B \implies A \cup B = B$

$A \cap B = A \implies A \subseteq B$

$A \subseteq B \implies A \cap B = A$

$A \cup B = A \implies B \subseteq A$

Question 19

$$\text{Let } B = {{a}, a, {a, b}}. \text{ Which statements are true? (Select all correct answers)}$$

${a} \in B$
$a \in B$
$b \in B$
${a} \subseteq B$
${a, b} \subseteq B$

[!Answer]-

${a} \in B$

$a \in B$

${a} \subseteq B$

Question 20

$$\text{How many students play either soccer or basketball?}$$

[!Answer]-

$27$

Question 21

$$\text{Let } A={1, {1}, {2, 3}, 4}. \text{ Which of the following are elements of A? (Select all correct answers)}$$

$1$
$3$
${1}$
${2, 3}$
$2$

[!Answer]-

$1$

${1}$

${2, 3}$

Question 22

$$\text{Let } C = {\emptyset, {\emptyset}}. \text{ Which are true? (Select all correct answers)}$$

$\emptyset \in C$
${\emptyset} \in C$
${\emptyset} \subseteq C$
${{\emptyset}} \subseteq C$

[!Answer]-

$\emptyset \in C$

${\emptyset} \in C$

${\emptyset} \subseteq C$

${{\emptyset}} \subseteq C$

Question 23

$$\text{If } A \cap B = \emptyset, \text{ which must be true? (Select all that are correct)}$$

$A - B = A$
$A = \emptyset \text{ or } B = \emptyset$
$A \subset B$
$B - A = B$

[!Answer]-

$A - B = A$

$B - A = B$

Question 24

$$\text{Let } |A| = 12, |B| = 9, |A \cap B| = 4. \text{ Find } |A \cup B|$$

$21$
$17$
$25$
$5$

[!Answer]-

$17$

Question 25

$$\text{Let } A={1, 2}. \text{ Which are elements of } A \times A? \text{ (Select all that are correct)}$$

$(1, 1)$
$(2, 1)$
${1, 2}$
$(1, 2)$
$(2, 2)$

[!Answer]-

$(1, 1)$

$(2, 1)$

$(1, 2)$

$(2, 2)$

Question 26

$$\text{If } |U| = 40, |A| = 18, |B| = 20, |A \cap B| = 5, \text{ then } |(A \cup B)^C| = ?$$

[!Answer]-

$7$

Question 27

$$\text{If } A \subseteq B \text{ and } B \subseteq C, \text{ which must be true?}$$

$A \subseteq C$
$C \subseteq A$
$A = C$
$A \subset C$

[!Answer]-

$A \subseteq C$

Question 28

$$(A \cap B) \cup A = A$$

True
False

[!Answer]-

True

Question 29

$$\text{Let } A={1, 2, 3} \text{ and } B={1, 2, 3, 4}. \text{ Which are true (Select all correct answers)}$$

$A \subset B$
$A = B$
$A \subseteq A$
$A \subseteq B$
$B \subseteq A$

[!Answer]-

$A \subset B$

$A \subseteq A$

$A \subseteq B$