Venn Diagrams — Hard Practice Quiz

A Set Theory cheat sheet for Venn Diagrams — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

AB
Union A ∪ B — shaded: everything in either set
$$A \cup B$$
AB
Intersection A ∩ B — shaded: only the overlap
$$A \cap B$$
AB
Difference A minus B — shaded: in A but not B
$$A \setminus B$$
AU
Complement Aᶜ — shaded: everything outside A within U
$$A^{c}$$
AB
Symmetric difference A △ B — shaded: in exactly one set
$$A \triangle B$$
AB
Subset A ⊆ B — A sits entirely inside B
$$A \subseteq B$$
AB
Disjoint A ∩ B = ∅ — no overlap
$$A \cap B = \varnothing$$

Practice quiz

  1. Which of the following expressions is equivalent to $A \setminus (B \cup C)$?

    • $A \cap B^c \cap C^c$
    • $A \setminus B \cup A \setminus C$
    • $(A \setminus B) \cup C^c$
    • $(A \cap B^c) \cup (A \cap C^c)$

    Answer: $A \cap B^c \cap C^c$

  2. Given a universal set $U$, if $A$ and $B$ are two subsets of $U$, which expression is equivalent to $(A^c \cup B^c)^c$?

    • $A \cap B$
    • $A \cup B$
    • $A^c \cap B^c$
    • $A \setminus B$

    Answer: $A \cap B$

  3. The symmetric difference $A \triangle B$ can be expressed in terms of union and intersection. Which of the following is a correct representation?

    • $(A \cup B) \setminus (A \cap B)$
    • $(A \cap B) \setminus (A \cup B)$
    • $A \cup B \cup (A \cap B)$
    • $A \cap B \cap (A \cup B)^c$

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

  4. If $A \subseteq B$, what is the simplified form of $(A \cup B) \cap A^c$?

    • $B \setminus A$
    • $A$
    • $B$
    • $\varnothing$

    Answer: $B \setminus A$

  5. Consider three sets $A$, $B$, and $C$. If $A \cap B = \varnothing$ and $B \subseteq C$, which of the following statements must be true?

    • $A \setminus B = A$
    • $A \cap C = \varnothing$
    • $A \subseteq C$
    • $C \setminus A = C$

    Answer: $A \setminus B = A$

  6. Which of the following expressions is equivalent to $(A \setminus B) \cup (B \setminus A)$?

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

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

  7. If $A$ and $B$ are two sets, simplify the expression $(A \cap B^c)^c \cup B$.

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

    Answer: $A^c \cup B$

  8. Given sets $A$, $B$, and $C$. If $A \subseteq B$ and $B \cap C = \varnothing$, what can be concluded about $A \cap C$?

    • $A \cap C = \varnothing$
    • $A \cap C = A$
    • $A \cap C = C$
    • $A \cap C = B$

    Answer: $A \cap C = \varnothing$

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

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

    Answer: $A$

  10. Which of the following statements is always true for any sets $A$ and $B$?

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

    Answer: $A \triangle B = (A \cup B) \cap (A \cap B)^c$

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