Set Operations — Practice Quiz

A Set Theory cheat sheet for Set Operations — every key formula with its symbols defined — plus a medium-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 sets $A = \{ \text{p, q, r, s} \}$ and $B = \{ \text{r, s, t, u} \}$, what is the union $A \cup B$?

    • $\{ \text{p, q, r, s, t, u} \}$
    • $\{ \text{r, s} \}$
    • $\{ \text{p, q} \}$
    • $\{ \text{t, u} \}$

    Answer: $\{ \text{p, q, r, s, t, u} \}$

  2. Let $X = \{ 1, 3, 5, 7, 9 \}$ and $Y = \{ 2, 3, 5, 8, 9 \}$. Find the intersection $X \cap Y$.

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

    Answer: $\{ 3, 5, 9 \}$

  3. If $M = \{ \text{apple, banana, orange} \}$ and $N = \{ \text{banana, grape, kiwi} \}$, what is the set difference $M \setminus N$?

    • $\{ \text{apple, orange} \}$
    • $\{ \text{banana} \}$
    • $\{ \text{grape, kiwi} \}$
    • $\{ \text{apple, banana, orange, grape, kiwi} \}$

    Answer: $\{ \text{apple, orange} \}$

  4. Consider the universal set $U = \{ \text{a, b, c, d, e, f} \}$ and set $C = \{ \text{a, c, e} \}$. What is the complement $C^c$?

    • $\{ \text{a, c, e} \}$
    • $\{ \text{b, d, f} \}$
    • $\{ \text{a, b, c, d, e, f} \}$
    • $\{ \text{ } \}$

    Answer: $\{ \text{b, d, f} \}$

  5. Given sets $R = \{ \text{cat, dog, fish} \}$ and $S = \{ \text{dog, bird, hamster} \}$, calculate the symmetric difference $R \triangle S$.

    • $\{ \text{dog} \}$
    • $\{ \text{cat, fish, bird, hamster} \}$
    • $\{ \text{cat, dog, fish, bird, hamster} \}$
    • $\{ \text{ } \}$

    Answer: $\{ \text{cat, fish, bird, hamster} \}$

  6. Which of the following pairs of sets are disjoint?

    • $A = \{ 1, 2, 3 \}$ and $B = \{ 3, 4, 5 \}$
    • $C = \{ \text{red, blue} \}$ and $D = \{ \text{green, yellow} \}$
    • $E = \{ \text{even numbers} \}$ and $F = \{ \text{multiples of 4} \}$
    • $G = \{ \text{prime numbers} \}$ and $H = \{ \text{odd numbers} \}$

    Answer: $C = \{ \text{red, blue} \}$ and $D = \{ \text{green, yellow} \}$

  7. Let $P = \{ \text{heads, tails} \}$ and $Q = \{ \text{1, 2} \}$. What is the Cartesian product $P \times Q$?

    • $\{ (\text{heads, 1}), (\text{heads, 2}), (\text{tails, 1}), (\text{tails, 2}) \}$
    • $\{ (\text{heads, tails}), (1, 2) \}$
    • $\{ \text{heads, tails, 1, 2} \}$
    • $\{ (1, \text{heads}), (2, \text{heads}), (1, \text{tails}), (2, \text{tails}) \}$

    Answer: $\{ (\text{heads, 1}), (\text{heads, 2}), (\text{tails, 1}), (\text{tails, 2}) \}$

  8. For the set $K = \{ \alpha, \beta, \gamma \}$, how many elements are in its power set $\mathcal{P}(K)$?

    • $3$
    • $6$
    • $8$
    • $9$

    Answer: $8$

  9. Given $U = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$, $A = \{ 1, 2, 4, 5 \}$, and $B = \{ 2, 5, 6, 7 \}$. Find $(A \cap B)^c$.

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

    Answer: $\{ 1, 3, 4, 6, 7, 8 \}$

  10. Let $S_1 = \{ \text{a, b, c, d} \}$, $S_2 = \{ \text{c, d, e, f} \}$, and $S_3 = \{ \text{a, e, g} \}$. What is $(S_1 \setminus S_2) \cup S_3$?

    • $\{ \text{a, b, e, g} \}$
    • $\{ \text{a, b, c, d, e, f, g} \}$
    • $\{ \text{c, d} \}$
    • $\{ \text{a, b} \}$

    Answer: $\{ \text{a, b, e, g} \}$

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