Sets-Relations-And-Functions Question 123
Question: Let R be a set of all distinct numbers of the form $\frac{m}{n}$, where $m, n \in \{1,2,3,4,5\}$. What is the cardinality of the set R?
Options:
A) 15
B) 18
C) 21
D) 25
Correct Answer: C First, we need to list all possible fractions $\frac{m}{n}$ where $m, n \in \{1,2,3,4,5\}$: $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ $\frac{2}{1}, \frac{2}{2}, \frac{2}{3}, \frac{2}{4}, \frac{2}{5}$ $\frac{3}{1}, \frac{3}{2}, \frac{3}{3}, \frac{3}{4}, \frac{3}{5}$ $\frac{4}{1}, \frac{4}{2}, \frac{4}{3}, \frac{4}{4}, \frac{4}{5}$ $\frac{5}{1}, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}, \frac{5}{5}$ The total number of these fractions is $5 \times 5 = 25$. However, the question asks for distinct numbers, so we need to simplify these fractions and count only the unique ones. After simplification: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$ $2, 1, \frac{2}{3}, \frac{1}{2}, \frac{2}{5}$ $3, \frac{3}{2}, 1, \frac{3}{4}, \frac{3}{5}$ $4, 2, \frac{4}{3}, 1, \frac{4}{5}$ $5, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}, 1$ Counting the unique numbers: $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, 2, \frac{2}{3}, \frac{2}{5}, 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{5}, 4, \frac{4}{3}, \frac{4}{5}, 5, \frac{5}{2}, \frac{5}{3}, \frac{5}{4}$ Therefore, the cardinality of set R is 21.Show Answer
Answer:
Solution:
BETA
