Set Operations — Hard Practice Quiz

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

Formulas & key concepts

Union: everything in A or B (or both). Example: {1,2,3} ∪ {3,4} = {1,2,3,4}

$$A \cup B = \{\, x \mid x \in A \text{ or } x \in B \,\}$$

Intersection: the elements common to both. Example: {1,2,3} ∩ {2,3,4} = {2,3}

$$A \cap B = \{\, x \mid x \in A \text{ and } x \in B \,\}$$

Difference (A minus B): in A but not in B. Example: {1,2,3} minus {2} = {1,3}

$$A \setminus B = \{\, x \mid x \in A,\ x \notin B \,\}$$

Complement: everything in the universal set U that is outside A

$$A^{c} = U \setminus A = \{\, x \in U \mid x \notin A \,\}$$

Symmetric difference: in exactly one set, not both. Example: {1,2,3} △ {2,3,4} = {1,4}

$$A \triangle B = (A \setminus B) \cup (B \setminus A)$$

Disjoint sets share no elements at all

$$A \cap B = \varnothing$$

Cartesian product: all ordered pairs. Example: {1,2} × {x,y} = {(1,x),(1,y),(2,x),(2,y)}

$$A \times B = \{\, (a, b) \mid a \in A,\ b \in B \,\}$$

Power set: the set of all subsets of A, including the empty set and A itself

$$\mathcal{P}(A) = \{\, S \mid S \subseteq A \,\}$$

Practice quiz

  1. Given a universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$, set $A = \{1, 2, 3, 4\}$, and set $B = \{3, 4, 5, 6\}$. Which of the following represents $(A \cup B)^c$?

    • $\{7, 8\}$
    • $\{1, 2, 3, 4, 5, 6\}$
    • $\{1, 2, 7, 8\}$
    • $\{3, 4\}$

    Answer: $\{7, 8\}$

  2. Let $U = \{x \mid x \text{ is an integer and } 1 \le x \le 10\}$, $A = \{1, 2, 3, 4, 5\}$, and $B = \{4, 5, 6, 7\}$. Determine the set $(A \triangle B)^c$.

    • $\{1, 2, 3, 6, 7\}$
    • $\{4, 5, 8, 9, 10\}$
    • $\{8, 9, 10\}$
    • $\{4, 5\}$

    Answer: $\{4, 5, 8, 9, 10\}$

  3. Given sets $A = \{p, q\}$, $B = \{q, r\}$, and $C = \{s, t\}$. What is the Cartesian product $(A \setminus B) \times C$?

    • $\{\, (p, s), (p, t) \,\}$
    • $\{\, (q, s), (q, t) \,\}$
    • $\{\, (p, s), (q, t) \,\}$
    • $\{\, (p, q), (s, t) \,\}$

    Answer: $\{\, (p, s), (p, t) \,\}$

  4. Let $A = \{x \mid x \text{ is an even integer and } 1 \le x \le 6\}$ and $B = \{x \mid x \text{ is a prime number and } 1 \le x \le 7\}$. Find $\mathcal{P}(A \cap B)$.

    • $\{\varnothing, \{2\}\}$
    • $\{\varnothing, \{2\}, \{4\}, \{6\}\}$
    • $\{\varnothing, \{2\}, \{3\}, \{5\}, \{7\}\}$
    • $\{\varnothing, \{2\}, \{4\}, \{6\}, \{3\}, \{5\}, \{7\}\}$

    Answer: $\{\varnothing, \{2\}\}$

  5. Which of the following expressions is equivalent to the symmetric difference $A \triangle B$?

    • $(A \cup B) \setminus (A \cap B)$
    • $(A \cap B) \setminus (A \cup B)$
    • $A \cup B$
    • $A \cap B$

    Answer: $(A \cup B) \setminus (A \cap B)$

  6. If sets $A$ and $B$ are disjoint, which of the following statements is always true?

    • $A \cup B = U$
    • $A \subseteq B^c$
    • $A \triangle B = \varnothing$
    • $A^c \cap B^c = \varnothing$

    Answer: $A \subseteq B^c$

  7. If set $X$ has $m$ elements and set $Y$ has $n$ elements, what is the cardinality of the power set of their Cartesian product, i.e., $|\mathcal{P}(X \times Y)|$?

    • $2^{m+n}$
    • $2^{mn}$
    • $mn$
    • $m+n$

    Answer: $2^{mn}$

  8. Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, $A = \{1, 2, 3, 4, 5\}$, $B = \{4, 5, 6, 7\}$, and $C = \{1, 5, 8, 9\}$. Determine the set $((A \cup B) \setminus C)^c$.

    • $\{1, 5, 8, 9, 10\}$
    • $\{2, 3, 4, 6, 7\}$
    • $\{1, 2, 3, 4, 5, 6, 7\}$
    • $\{8, 9, 10\}$

    Answer: $\{1, 5, 8, 9, 10\}$

  9. Under what condition is the symmetric difference $A \triangle B$ equal to the union $A \cup B$?

    • When $A \subseteq B$
    • When $B \subseteq A$
    • When $A$ and $B$ are disjoint
    • When $A = B$

    Answer: When $A$ and $B$ are disjoint

  10. Given sets $A = \{a, b\}$ and $B = \{b, c\}$. Find $\mathcal{P}(A \setminus B) \cap \mathcal{P}(B \setminus A)$.

    • $\{\varnothing\}$
    • $\{\varnothing, \{a\}, \{c\}\}$
    • $\{\varnothing, \{a\}\}$
    • $\{\varnothing, \{c\}\}$

    Answer: $\{\varnothing\}$

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