Question

Let $S$ be a set of all the distinct numbers of the form $\dfrac{P}{Q}$, where $p,q\in \{1,2,3,4,5,6\}$. What is the cardinality of the set $S$?
A. $21$
B. $23$
C. $32$
D. $36$

Answer 1

Hint: To solve this question we will take each of the possible value of $p$and $q$ from $\{1,2,3,4,5,6\}$and determine all the numbers of the form $\dfrac{P}{Q}$ . We will first find all the fractional numbers which will be same and consider them only once so they won’t repeat thus only taking co-prime numbers as the value of $p$and $q$. After finding all the possible numbers we will add them and determine the cardinality of the set $S$.

Complete step by step solution:
 We have given a set $S$ which is the collections of all the distinct numbers in the form of $\dfrac{P}{Q}$, where $p,q\in \{1,2,3,4,5,6\}$ and we have to find the cardinality of the set $S$.
We know that cardinality of set means total number of the elements it will contain in its sample space. Now in set $S$ the numbers are of the form $\dfrac{P}{Q}$ where $p,q\in \{1,2,3,4,5,6\}$ . Because $\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{3}{6}$ and $\dfrac{1}{1}=\dfrac{2}{2}=\dfrac{3}{3}=\dfrac{4}{4}=\dfrac{5}{5}=\dfrac{6}{6}$ will be same we will only find co-prime numbers.
When $p=1,\,q=\{1,2,3,4,5,6\}$, the total number of elements in the set $S$will be $6$ that is $\dfrac{1}{1},\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{5},\dfrac{1}{6}$.
When $p=2,\,q=\{1,3,5\}$, the total number of elements in the set $S$will be $3$ that is $\dfrac{2}{1},\dfrac{2}{3},\dfrac{2}{5}$.
When $p=3,\,q=\{1,2,4,5\}$, the total number of elements in the set $S$will be $4$ that is $\dfrac{3}{1},\dfrac{3}{2},\dfrac{3}{4},\dfrac{3}{5}$.
When $p=4,\,q=\{1,3,5\}$, the total number of elements in the set $S$will be $3$ that is $\dfrac{4}{1},\dfrac{4}{3},\dfrac{4}{5}$.
When $p=5,\,q=\{1,2,3,4,6\}$, the total number of elements in the set $S$will be $5$ that is $\dfrac{5}{1},\dfrac{5}{2},\dfrac{5}{3},\dfrac{5}{4},\dfrac{5}{6}$.
When $p=6,\,q=\{1,5\}$, the total number of elements in the set $S$will be $2$ that is $\dfrac{6}{1},\dfrac{6}{5}$.
Now the cardinality of the set will be,
 $\begin{align}
  & =6+3+4+3+5+2 \\
 & =23 \\
\end{align}$
The cardinality of the set $S$which is the set of all the distinct numbers of the form $\dfrac{P}{Q}$ where $p,q\in \{1,2,3,4,5,6\}$ is $23$.

Option ‘B’ is correct

Note: Co-prime numbers can be defined as the numbers which have no common factors except one.