Set Basics & Notation — Practice Quiz

A Set Theory cheat sheet for Set Basics & Notation — every key formula with its symbols defined — plus a medium-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. Given $A = \{1, 3, 5, 7\}$, which of the following statements is true?

    • $2 \in A$
    • $5 \notin A$
    • $7 \in A$
    • $10 \in A$

    Answer: $7 \in A$

  2. Which set is equivalent to $A = \{ x \mid x \text{ is an even integer and } 1 < x < 9 \}$?

    • $A = \{2, 4, 6, 8\}$
    • $A = \{1, 2, 3, ..., 9\}$
    • $A = \{2, 4, 6\}$
    • $A = \{x \mid x = 2n, n \in \mathbb{Z}\}$

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

  3. Which of the following correctly describes the empty set?

    • A set containing the element $0$.
    • A set that contains all integers.
    • The set containing no elements.
    • A set that is not a subset of any other set.

    Answer: The set containing no elements.

  4. Given $P = \{a, b, c\}$ and $Q = \{a, b, c, d, e\}$, which statement is true?

    • $Q \subseteq P$
    • $P \subseteq Q$
    • $P = Q$
    • $d \in P$

    Answer: $P \subseteq Q$

  5. Let $X = \{1, 2, 3\}$ and $Y = \{1, 2, 3, 4\}$. Which of the following is true?

    • $Y \subset X$
    • $X = Y$
    • $X \subset Y$
    • $4 \notin Y$

    Answer: $X \subset Y$

  6. If $A = \{x \mid x \text{ is a prime number less than } 10\}$ and $B = \{2, 3, 5, 7\}$, then:

    • $A \subset B$
    • $B \subset A$
    • $A = B$
    • $|A| \neq |B|$

    Answer: $A = B$

  7. What is the cardinality of the set $S = \{x \mid x \text{ is an integer and } -2 < x \le 3\}$?

    • $|S| = 4$
    • $|S| = 5$
    • $|S| = 6$
    • $|S| = 3$

    Answer: $|S| = 5$

  8. Given $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4, 5\}$. Which statement is true?

    • $A \subset B \text{ and } |A| = |B|$
    • $A \subseteq B \text{ and } |A| < |B|$
    • $B \subseteq A \text{ and } |B| > |A|$
    • $A = B \text{ and } |A| = 3$

    Answer: $A \subseteq B \text{ and } |A| < |B|$

  9. Consider the set $C = \{ y \mid y = n^2, n \in \mathbb{N} \text{ and } n < 5 \}$. Which of the following elements belongs to $C$? (Assume $\mathbb{N} = \{1, 2, 3, ...\}$)

    • $0$
    • $9$
    • $16$
    • $25$

    Answer: $9$

  10. Let $M = \{x, y, z\}$ and $N = \{z, y, x\}$. Which statement is true?

    • $M \subset N$
    • $N \subset M$
    • $M = N$
    • $|M| \neq |N|$

    Answer: $M = N$

Select a subject

Select a subject from the left panel to begin exploring formulas.