Algebra of Sets (Laws) — Hard Practice Quiz

A Set Theory cheat sheet for Algebra of Sets (Laws) — every key formula with its symbols defined — plus a hard-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. Simplify the set expression $( (A \cap B)^{c} \cup A )^{c}$.

    • $\varnothing$
    • $A$
    • $B$
    • $A^{c}$

    Answer: $\varnothing$

  2. Simplify the set expression $A \cap (B \cup A^{c})$.

    • $A \cup B$
    • $A \cap B$
    • $A$
    • $A^{c}$

    Answer: $A \cap B$

  3. Simplify the set expression $(A \cup B^{c})^{c} \cup (A^{c} \cap B^{c})$.

    • $A$
    • $B$
    • $A^{c}$
    • $U$

    Answer: $A^{c}$

  4. Given that $A \subseteq B$, simplify the set expression $(A \cup B^{c}) \cap B$.

    • $A$
    • $B$
    • $\varnothing$
    • $U$

    Answer: $A$

  5. Which of the following expressions is equivalent to $A \cup (A^{c} \cap B)$?

    • $A \cap B$
    • $A$
    • $B$
    • $A \cup B$

    Answer: $A \cup B$

  6. Simplify the set expression $(A \cap B) \cup (A \cap B^{c}) \cup (A^{c} \cap B)$.

    • $A$
    • $B$
    • $A \cup B$
    • $A \cap B$

    Answer: $A \cup B$

  7. If $A \cap B = \varnothing$ and $A \cup B = U$, what is the relationship between sets $A$ and $B$?

    • $A = B$
    • $A \subseteq B$
    • $B = A^{c}$
    • $A = \varnothing$

    Answer: $B = A^{c}$

  8. Simplify the set expression $(A \cup B) \cap (A \cup B^{c}) \cap (A^{c} \cup B)$.

    • $A$
    • $B$
    • $A \cup B$
    • $A \cap B$

    Answer: $A \cap B$

  9. Simplify the set expression $( (A \cup B^{c})^{c} \cup (A^{c} \cap B) )^{c}$.

    • $A \cap B$
    • $A^{c} \cup B$
    • $A \cup B^{c}$
    • $A^{c} \cap B^{c}$

    Answer: $A \cup B^{c}$

  10. Given that $A \cap B = \varnothing$ and $A \cup C = U$, simplify the set expression $(A \cup B) \cap (A^{c} \cup C)$.

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

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

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