Set Basics & Notation — Hard Practice Quiz

A Set Theory cheat sheet for Set Basics & Notation — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Element of: x is a member of set A

$$x \in A$$

Not an element: x is not in set A

$$x \notin A$$

Roster notation: list every element between braces

$$A = \{2, 4, 6, 8\}$$

Set-builder notation: read the bar as "such that". This is the set of even naturals

$$A = \{\, x \mid x = 2n,\ n \in \mathbb{N} \,\}$$

Empty set: the unique set with no elements

$$\varnothing = \{\ \}$$

Subset: every element of A is also in B

$$A \subseteq B$$

Proper subset: A is inside B and A is not equal to B

$$A \subset B$$

Set equality: two sets are equal exactly when each contains the other

$$A = B \iff A \subseteq B \text{ and } B \subseteq A$$

Cardinality: the number of elements in a set

$$|A| = 4 \text{ for } A = \{2, 4, 6, 8\}$$

Practice quiz

  1. Let $S = \{\, x \mid x \in \mathbb{N} \text{ and } x \text{ is a prime number less than } 10 \,\}$ and $T = \{2, 3, 5, 7, 9\}$. Which of the following statements is true?

    • A) $S = T$
    • B) $9 \in S$
    • C) $S \subseteq T$
    • D) $T \subseteq S$

    Answer: C) $S \subseteq T$

  2. Consider set $A = \{\, n \mid n \in \mathbb{Z} \text{ and } n^2 < 10 \,\}$ and set $B = \{-3, -2, -1, 0, 1, 2, 3\}$. Which statement accurately describes the relationship between $A$ and $B$?

    • A) $A \subset B$ because $|A| < |B|$
    • B) $A = B$ because $A \subseteq B$ and $B \subseteq A$
    • C) $A \not\subseteq B$ because $-3 \notin B$
    • D) $|A| \ne |B|$ because $A$ is defined by a condition

    Answer: B) $A = B$ because $A \subseteq B$ and $B \subseteq A$

  3. Let $X = \{\, k \mid k \in \mathbb{N} \text{ and } k^2 = -1 \,\}$. Which of the following statements is true regarding $X$?

    • A) $X \subset \{1, 2, 3\}$
    • B) $|X| = 1$
    • C) $1 \in X$
    • D) $X = \{0\}$

    Answer: A) $X \subset \{1, 2, 3\}$

  4. Given $P = \{1, 3, 5, 7, 9\}$ and $Q = \{\, y \mid y \text{ is an odd natural number less than } 10 \,\}$. Determine the relationship between $P$ and $Q$.

    • A) $P \subset Q$
    • B) $Q \subset P$
    • C) $P = Q$
    • D) $P \notin Q$

    Answer: C) $P = Q$

  5. Let $D = \{\, m \mid m \in \mathbb{Z} \text{ and } -2 \le m < 3 \,\}$. What is the cardinality of $D$?

    • A) $3$
    • B) $4$
    • C) $5$
    • D) $6$

    Answer: C) $5$

  6. Given $A = \{x, y, z\}$, $B = \{z, y, x\}$, and $C = \{\, k \mid k \in \mathbb{N} \text{ and } k < 1 \,\}$. Which of the following statements is false?

    • A) $A \subseteq B$
    • B) $C \subset A$
    • C) $|A| = |B|$
    • D) $A \subset B$

    Answer: D) $A \subset B$

  7. If $M$ and $N$ are two sets such that $M \subseteq N$ and $N \subseteq M$, which of the following must be true?

    • A) $M \subset N$
    • B) $M = N$
    • C) $|M| \ne |N|$
    • D) $M \ne N$

    Answer: B) $M = N$

  8. Let $F = \{\, p \mid p \in \mathbb{Z} \text{ and } p \text{ is a factor of } 18 \text{ and } p > 0 \,\}$. What is the cardinality of $F$?

    • A) $4$
    • B) $6$
    • C) $8$
    • D) $12$

    Answer: B) $6$

  9. Which of the following statements is always true for any set $S$?

    • A) $S \subset S$
    • B) $\varnothing \subset S$
    • C) $\varnothing \subseteq S$
    • D) $S \subseteq \varnothing$

    Answer: C) $\varnothing \subseteq S$

  10. Let $G = \{\, k \mid k \in \mathbb{N} \text{ and } k \text{ is a multiple of } 3 \text{ and } k < 15 \,\}$. What is the cardinality of $G$?

    • A) $3$
    • B) $4$
    • C) $5$
    • D) $14$

    Answer: B) $4$

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