Algebra of Sets (Laws) — Practice Quiz

A Set Theory cheat sheet for Algebra of Sets (Laws) — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Commutative laws: order does not matter

$$A \cup B = B \cup A \qquad A \cap B = B \cap A$$

Associative law: grouping does not matter (the same holds for intersection)

$$(A \cup B) \cup C = A \cup (B \cup C)$$

Distributive law: intersection distributes over union

$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$

Distributive law: union distributes over intersection

$$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$

Identity laws

$$A \cup \varnothing = A \qquad A \cap U = A$$

Domination (null) laws

$$A \cup U = U \qquad A \cap \varnothing = \varnothing$$

Idempotent laws

$$A \cup A = A \qquad A \cap A = A$$

Complement laws

$$A \cup A^{c} = U \qquad A \cap A^{c} = \varnothing$$

Double complement (involution) law

$$(A^{c})^{c} = A$$

De Morgan law: the complement of a union is the intersection of the complements

$$(A \cup B)^{c} = A^{c} \cap B^{c}$$

De Morgan law: the complement of an intersection is the union of the complements

$$(A \cap B)^{c} = A^{c} \cup B^{c}$$

Absorption laws

$$A \cup (A \cap B) = A \qquad A \cap (A \cup B) = A$$

Practice quiz

  1. Which of the following statements correctly illustrates the commutative law for set union?

    • $A \cup B = B \cup A$
    • $A \cap B = B \cap A$
    • $(A \cup B) \cup C = A \cup (B \cup C)$
    • $A \cup \varnothing = A$

    Answer: $A \cup B = B \cup A$

  2. Apply the distributive law to expand the expression $X \cap (Y \cup Z)$.

    • $(X \cap Y) \cup (X \cap Z)$
    • $(X \cup Y) \cap (X \cup Z)$
    • $(X \cap Y) \cup Z$
    • $X \cup (Y \cap Z)$

    Answer: $(X \cap Y) \cup (X \cap Z)$

  3. According to De Morgan's laws, the complement of the union of two sets $P$ and $Q$, denoted as $(P \cup Q)^c$, is equivalent to which expression?

    • $P^c \cap Q^c$
    • $P^c \cup Q^c$
    • $(P \cap Q)^c$
    • $P \cap Q^c$

    Answer: $P^c \cap Q^c$

  4. Given a universal set $U$ and any set $S$, what is the result of $S \cap U$?

    • $S$
    • $U$
    • $\varnothing$
    • $S^c$

    Answer: $S$

  5. For any set $M$ and the universal set $U$, which of the following is true?

    • $M \cup U = U$
    • $M \cap U = U$
    • $M \cup \varnothing = U$
    • $M \cap \varnothing = M$

    Answer: $M \cup U = U$

  6. Which of the following expressions is an example of the absorption law?

    • $A \cup (A \cap B) = A$
    • $A \cup A = A$
    • $A \cup A^c = U$
    • $(A \cup B)^c = A^c \cap B^c$

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

  7. If $A$ is a set and $A^c$ is its complement, what is the intersection of $A$ and $A^c$?

    • $\varnothing$
    • $U$
    • $A$
    • $A^c$

    Answer: $\varnothing$

  8. Simplify the expression $(X^c)^c$.

    • $X$
    • $X^c$
    • $\varnothing$
    • $U$

    Answer: $X$

  9. Which law states that the grouping of sets in a union operation does not affect the result, i.e., $(A \cup B) \cup C = A \cup (B \cup C)$?

    • Associative law
    • Commutative law
    • Distributive law
    • Idempotent law

    Answer: Associative law

  10. Which expression is equivalent to $P \cup (Q \cap R)$ according to the distributive law?

    • $(P \cup Q) \cap (P \cup R)$
    • $(P \cap Q) \cup (P \cap R)$
    • $(P \cup Q) \cap R$
    • $P \cap (Q \cup R)$

    Answer: $(P \cup Q) \cap (P \cup R)$

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