Cardinality & Counting — Practice Quiz

A Set Theory cheat sheet for Cardinality & Counting — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Inclusion–Exclusion (two sets): add the sizes, then subtract the overlap counted twice

$$|A \cup B| = |A| + |B| - |A \cap B|$$

Inclusion–Exclusion (three sets)

$$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$$

Sum rule for disjoint sets: with no overlap there is nothing to subtract

$$|A \cup B| = |A| + |B| \quad \text{if } A \cap B = \varnothing$$

Product rule: pairs from A and B. Example: 3 × 2 = 6 ordered pairs

$$|A \times B| = |A| \cdot |B|$$

Number of subsets: a set with n elements has 2^n subsets. Example: {a,b,c} has 8 subsets

$$|\mathcal{P}(A)| = 2^{|A|}$$

Practice quiz

  1. In a class of $30$ students, $18$ play soccer and $15$ play basketball. If $7$ students play both sports, how many students play at least one sport?

    • $26$
    • $33$
    • $23$
    • $40$

    Answer: $26$

  2. A survey of $100$ people showed that $60$ read newspaper A and $40$ read newspaper B. If $25$ people read neither, how many people read both newspaper A and B?

    • $15$
    • $20$
    • $25$
    • $30$

    Answer: $25$

  3. A restaurant offers $5$ different main courses and $3$ different desserts. If a customer chooses either a main course or a dessert, but not both, how many different choices are there?

    • $8$
    • $15$
    • $5$
    • $3$

    Answer: $8$

  4. A license plate consists of $3$ letters followed by $3$ digits. If repetition is allowed, how many different license plates are possible? (Assume $26$ letters and $10$ digits).

    • $26^3 + 10^3$
    • $26 \times 3 + 10 \times 3$
    • $26^3 \times 10^3$
    • $26 \times 10 \times 6$

    Answer: $26^3 \times 10^3$

  5. How many distinct subsets can be formed from the set $A = \{a, b, c, d, e\}$?

    • $5$
    • $10$
    • $25$
    • $32$

    Answer: $32$

  6. Which of the following formulas correctly represents the Inclusion-Exclusion Principle for three sets $A$, $B$, and $C$?

    • $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C|$
    • $|A \cup B \cup C| = |A| + |B| + |C| + |A \cap B| + |A \cap C| + |B \cap C| + |A \cap B \cap C|$
    • $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$
    • $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B \cap C|$

    Answer: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

  7. A student needs to choose one shirt from $4$ options, one pair of pants from $3$ options, and one pair of shoes from $2$ options. How many different outfits can the student create?

    • $9$
    • $12$
    • $24$
    • $48$

    Answer: $24$

  8. If a set has $128$ subsets, how many elements does the set contain?

    • $6$
    • $7$
    • $8$
    • $128$

    Answer: $7$

  9. In a group of $100$ students, $40$ study Math, $30$ study Physics, and $20$ study Chemistry. $10$ study Math and Physics, $8$ study Math and Chemistry, $5$ study Physics and Chemistry. If $3$ students study all three subjects, how many students study at least one subject?

    • $65$
    • $70$
    • $75$
    • $80$

    Answer: $70$

  10. A committee needs to select either a chairperson from $5$ candidates OR a secretary from $3$ candidates. How many ways can this selection be made?

    • $8$
    • $15$
    • $5$
    • $3$

    Answer: $8$

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