Self-Assessment Quiz: Sets

Twenty questions to check your understanding. Answer before opening the key. Aim for 16+.

Question 1

Which of the following equals $\{1, 2, 3\}$?

A) $\{1, 2, 3, 3\}$ B) $(1, 2, 3)$ C) $\{ \{1\}, \{2\}, \{3\} \}$ D) $\{1, 2\}$

Question 2

The cardinality $\lvert \{ \emptyset, \{\emptyset\}, \{1, 2\} \} \rvert$ is:

A) 0 B) 2 C) 3 D) 4

Question 3

For the set $A = \{a, \{b, c\}\}$, which statement is true?

A) $b \in A$ B) $\{b, c\} \in A$ C) $\{b, c\} \subseteq A$ D) $\lvert A \rvert = 3$

Question 4

The empty set $\emptyset$ is a subset of:

A) only itself B) only sets that contain $0$ C) every set D) no set, since it has nothing to contribute

Question 5

If $\lvert S \rvert = 5$, then $\lvert \mathcal{P}(S) \rvert$ equals:

A) 5 B) 10 C) 25 D) 32

Question 6

Which is an element of $\mathcal{P}(\{a, b\})$?

A) $a$ B) $\{a\}$ C) $(a, b)$ D) $b$

Question 7

With universe $U = \{1, \dots, 6\}$, $A = \{1, 2, 3\}$, $B = \{3, 4\}$, the set $A \setminus B$ is:

A) $\{1, 2\}$ B) $\{4\}$ C) $\{1, 2, 4\}$ D) $\{3\}$

Question 8

$A \cap B = \emptyset$ means $A$ and $B$ are:

A) equal B) complementary C) disjoint D) subsets of each other

Question 9

Which set operation corresponds to logical negation ($\neg$)?

A) union B) intersection C) complement D) Cartesian product

Question 10

The identity $\overline{A \cup B} = \overline{A} \cap \overline{B}$ is "secretly" which logical equivalence?

A) $\neg(p \land q) \equiv \neg p \lor \neg q$ B) $\neg(p \lor q) \equiv \neg p \land \neg q$ C) $p \rightarrow q \equiv \neg p \lor q$ D) $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$

Question 11

For $A = \{1, 2\}$ and $B = \{x, y, z\}$, $\lvert A \times B \rvert$ is:

A) 5 B) 6 C) 8 D) 9

Question 12

Which is true about ordered pairs and Cartesian products?

A) $(a, b) = (b, a)$ always B) $A \times B = B \times A$ always C) $(a, b) = (c, d)$ iff $a = c$ and $b = d$ D) $A \times \emptyset = A$

Question 13

To prove two sets $A$ and $B$ are equal by the element method, you show:

A) $A \cap B = \emptyset$ B) $\lvert A \rvert = \lvert B \rvert$ C) $A \subseteq B$ and $B \subseteq A$ D) $A \cup B = A$

Question 14 (True/False, justify)

True or false: A Venn diagram showing both sides of an identity shaded identically constitutes a proof of that identity. Justify in one sentence.

Question 15 (True/False, justify)

True or false: In Python, {} creates an empty set. Justify in one sentence.

Question 16

You need a set whose elements are themselves sets (for example, a power set in code). The inner sets must be:

A) list objects B) tuple objects C) frozenset objects D) dict objects

Question 17

The chief reason to choose a set over a list for membership testing is:

A) sets preserve insertion order B) x in s is $O(1)$ average for a set vs. $O(n)$ for a list C) sets allow duplicate elements D) sets support indexing like s[3]

Question 18 (True/False, justify)

True or false: $A \subseteq A$ for every set $A$, but $A \subset A$ for no set $A$. Justify in one sentence.

Question 19 (Short answer)

In one sentence, state the single logical equivalence that carries all the mathematical content of a set-identity proof by the element method, and name the "bookkeeping" steps that surround it.

Question 20 (Short answer)

A feature-flag system has 20 independent on/off flags. In one sentence, explain why there are exactly $2^{20}$ possible configurations, citing the relevant theorem from §8.2.

Answer Key

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