1. **Problem Statement:** Find the cardinality of the set $S$ of all distinct numbers of the form $\frac{p}{q}$ where $p, q \in \{1, 2, 3, 4, 5, 6\}$. 2. **Understanding the problem:** We want to find how many unique fractions can be formed with numerator and denominator from 1 to 6. 3. **Step 1: Total fractions:** There are $6$ choices for $p$ and $6$ choices for $q$, so total fractions are $6 \times 6 = 36$. 4. **Step 2: Remove duplicates:** Fractions like $\frac{2}{4}$ and $\frac{1}{2}$ represent the same number. We need to count only distinct simplified fractions. 5. **Step 3: Simplify fractions:** Each fraction $\frac{p}{q}$ can be reduced by their greatest common divisor (gcd). The distinct fractions correspond to all fractions in lowest terms with numerator and denominator between 1 and 6. 6. **Step 4: Count distinct fractions:** We count all fractions $\frac{p}{q}$ with $1 \leq p,q \leq 6$ and $\gcd(p,q) = 1$. 7. **Step 5: Calculate:** - For $q=1$, $p=1$ to $6$, all gcd=1, so 6 fractions. - For $q=2$, $p=1$ to $6$, gcd=1 for $p=1,3,5$ (3 fractions). - For $q=3$, gcd=1 for $p=1,2,4,5$ (4 fractions). - For $q=4$, gcd=1 for $p=1,3,5$ (3 fractions). - For $q=5$, gcd=1 for $p=1,2,3,4,6$ (5 fractions). - For $q=6$, gcd=1 for $p=1,5$ (2 fractions). Total distinct fractions = $6 + 3 + 4 + 3 + 5 + 2 = 23$. 8. **Final answer:** The cardinality of $S$ is $23$. **Answer:** (b) 23