Cardinality & Counting — Hard Practice Quiz

A Set Theory cheat sheet for Cardinality & Counting — every key formula with its symbols defined — plus a hard-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. Set $A$ has $32$ subsets, and set $B$ has $64$ subsets. If the intersection $A \cap B$ has $8$ subsets, what is the cardinality of the union $A \cup B$?

    • $8$
    • $11$
    • $14$
    • $88$

    Answer: $8$

  2. Given two sets $A$ and $B$, if the cardinality of their Cartesian product is $|A \times B| = 60$, and the cardinality of set $A$ is $|A| = 10$, and the cardinality of their intersection is $|A \cap B| = 3$, what is the cardinality of their union $|A \cup B|$?

    • $13$
    • $16$
    • $19$
    • $57$

    Answer: $13$

  3. Set $A$ has $16$ subsets, set $B$ has $32$ subsets, and set $C$ has $8$ subsets. If $|A \cap B| = 2$, $|A \cap C| = 1$, $|B \cap C| = 1$, and $A \cap B \cap C = \varnothing$, what is the cardinality of $A \cup B \cup C$?

    • $8$
    • $12$
    • $16$
    • $52$

    Answer: $8$

  4. Given two sets $A$ and $B$, if $|A \times B| = 40$, $|A \cup B| = 13$, and $|A \cap B| = 2$, what is the cardinality of the larger set between $A$ and $B$?

    • $5$
    • $8$
    • $13$
    • $40$

    Answer: $8$

  5. Set $A$ has $8$ subsets and set $B$ has $16$ subsets. If $A$ and $B$ are disjoint, how many subsets does their union $A \cup B$ have?

    • $7$
    • $24$
    • $128$
    • $4096$

    Answer: $128$

  6. If $|A \cup B| = 12$, $|A \cap B| = 3$, and set $A$ has $64$ subsets, what is the cardinality of the Cartesian product $B \times A$?

    • $18$
    • $54$
    • $72$
    • $15$

    Answer: $54$

  7. Let $|A|=5$, $|B|=7$, and $|C|=6$. If set $A$ is disjoint from both set $B$ and set $C$, but $|B \cap C|=2$, what is the cardinality of $A \cup B \cup C$?

    • $16$
    • $18$
    • $20$
    • $14$

    Answer: $16$

  8. Set $A$ has $256$ subsets. If $|A \cup B| = 15$ and $|A \cap B| = 3$, what is the cardinality of the Cartesian product $A \times B$?

    • $18$
    • $45$
    • $80$
    • $2560$

    Answer: $80$

  9. In a group of $100$ people, $50$ like coffee ($C$), $40$ like tea ($T$), and $30$ like juice ($J$). $20$ like coffee and tea, $15$ like coffee and juice, and $10$ like tea and juice. If $80$ people like at least one of these beverages, how many people like all three beverages?

    • $0$
    • $5$
    • $10$
    • $15$

    Answer: $5$

  10. Set $A$ has $16$ subsets. If the cardinality of the Cartesian product $|A \times B| = 28$ and the cardinality of the union $|A \cup B| = 10$, what is the cardinality of the intersection $A \cap B$?

    • $0$
    • $1$
    • $2$
    • $3$

    Answer: $1$

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