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 represents the Commutative Law for set union?

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

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

  2. According to the Distributive Law, $A \cap (B \cup C)$ is equivalent to:

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

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

  3. The expression $(A \cup B)^{c}$ is equivalent to which of the following by De Morgan's Law?

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

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

  4. Which law states that the union of any set $A$ with the empty set $\varnothing$ is $A$?

    • Domination Law
    • Idempotent Law
    • Identity Law
    • Complement Law

    Answer: Identity Law

  5. Simplify the expression $A \cup (A \cap B)$ using set laws.

    • $A$
    • $B$
    • $A \cap B$
    • $U$ (Universal Set)

    Answer: $A$

  6. What is the result of $A \cap A^{c}$?

    • $U$ (Universal Set)
    • $\varnothing$ (Empty Set)
    • $A$
    • $A^{c}$

    Answer: $\varnothing$ (Empty Set)

  7. The Double Complement Law states that $(A^{c})^{c}$ is equal to:

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

    Answer: $A$

  8. Which law allows us to write $(A \cup B) \cup C$ as $A \cup (B \cup C)$?

    • Commutative Law
    • Distributive Law
    • Associative Law
    • Idempotent Law

    Answer: Associative Law

  9. If $U$ represents the universal set, what is $A \cup U$?

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

    Answer: $U$

  10. The expression $A \cap A$ simplifies to:

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

    Answer: $A$

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