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 $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, what is $A \cup B$?

    • $\{1, 2, 3, 4, 5\}$
    • $\{3\}$
    • $\{1, 2, 4, 5\}$
    • $\{1, 2, 3\}$

    Answer: $\{1, 2, 3, 4, 5\}$

  2. If $P = \{a, b, c, d\}$ and $Q = \{c, d, e, f\}$, what is $P \cap Q$?

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

    Answer: $\{c, d\}$

  3. For sets $X = \{\text{apple, banana, cherry}\}$ and $Y = \{\text{banana, date}\}$, what is $X \setminus Y$?

    • $\{\text{apple, cherry}\}$
    • $\{\text{banana}\}$
    • $\{\text{apple, banana, cherry, date}\}$
    • $\{\text{date}\}$

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

  4. Let the universal set be $U = \{1, 2, 3, 4, 5\}$ and set $A = \{2, 4\}$. What is the complement of $A$, denoted as $A^c$?

    • $\{1, 3, 5\}$
    • $\{2, 4\}$
    • $\{1, 2, 3, 4, 5\}$
    • $\varnothing$

    Answer: $\{1, 3, 5\}$

  5. Given sets $M = \{\text{red, blue, green}\}$ and $N = \{\text{blue, yellow}\}$, what is the symmetric difference $M \triangle N$?

    • $\{\text{red, green, yellow}\}$
    • $\{\text{blue}\}$
    • $\{\text{red, blue, green, yellow}\}$
    • $\varnothing$

    Answer: $\{\text{red, green, yellow}\}$

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

    • $A=\{1,2\}, B=\{2,3\}$
    • $C=\{\text{a,b}\}, D=\{\text{c,d}\}$
    • $E=\{\text{x,y}\}, F=\{\text{y,z}\}$
    • $G=\{\text{p,q}\}, H=\{\text{q,r}\}$

    Answer: $C=\{\text{a,b}\}, D=\{\text{c,d}\}$

  7. If $A = \{\text{H, T}\}$ and $B = \{1, 2\}$, what is the Cartesian product $A \times B$?

    • $\{(\text{H},1), (\text{H},2), (\text{T},1), (\text{T},2)\}$
    • $\{(\text{H},1), (\text{T},2)\}$
    • $\{\text{H, T, 1, 2}\}$
    • $\{(1,\text{H}), (2,\text{H}), (1,\text{T}), (2,\text{T})\}$

    Answer: $\{(\text{H},1), (\text{H},2), (\text{T},1), (\text{T},2)\}$

  8. What is the power set $\mathcal{P}(S)$ for the set $S = \{\text{a, b}\}$?

    • $\{\varnothing, \{\text{a}\}, \{\text{b}\}, \{\text{a, b}\}\}$
    • $\{\{\text{a}\}, \{\text{b}\}\}$
    • $\{\{\text{a, b}\}\}$
    • $\varnothing$

    Answer: $\{\varnothing, \{\text{a}\}, \{\text{b}\}, \{\text{a, b}\}\}$

  9. Given sets $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$, and $C = \{3, 4, 5\}$, find $(A \cup B) \cap C$.

    • $\{3, 4\}$
    • $\{1, 2, 3, 4, 5\}$
    • $\{2, 3\}$
    • $\{3\}$

    Answer: $\{3, 4\}$

  10. Let the universal set be $U = \{1, 2, 3, 4, 5, 6\}$, set $A = \{1, 3, 5\}$, and set $B = \{2, 3, 6\}$. Find $A \setminus B^c$.

    • $\{3\}$
    • $\{1, 5\}$
    • $\{2, 6\}$
    • $\{1, 3, 5\}$

    Answer: $\{3\}$

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