Let S be a set of all distinct numbers of the form \(\frac{{\rm{p}}}{{\rm{q}}}\) , where p, q ∈ {1, 2, 3, 4, 5, 6}. What is the the cardinality of the set S?
23
Finding the Cardinality of a Set of Distinct Fractions
The problem asks us to determine the number of unique elements in a set S. The set S contains fractions of the form \(\frac{p}{q}\), where both p and q are chosen from the set of integers {1, 2, 3, 4, 5, 6}. The cardinality of the set S is the total count of these distinct fractions.
The possible values for p are {1, 2, 3, 4, 5, 6}.
The possible values for q are {1, 2, 3, 4, 5, 6}.
To form the fractions \(\frac{p}{q}\), we can combine any value of p with any value of q. Since there are 6 choices for p and 6 choices for q, there are a total of \(6 \times 6 = 36\) possible fractions initially. However, the set S contains only the distinct numbers, meaning we must eliminate any duplicate values.
Listing all Possible Fractions \(\frac{p}{q}\)
Let's systematically list all 36 possible fractions by combining each value of p with each value of q and then simplify them to identify distinct values.
| p | q | Fraction \(\frac{p}{q}\) | Simplified Fraction |
|---|---|---|---|
| 1 | 1 | \(\frac{1}{1}\) | 1 |
| 1 | 2 | \(\frac{1}{2}\) | \(\frac{1}{2}\) |
| 1 | 3 | \(\frac{1}{3}\) | \(\frac{1}{3}\) |
| 1 | 4 | \(\frac{1}{4}\) | \(\frac{1}{4}\) |
| 1 | 5 | \(\frac{1}{5}\) | \(\frac{1}{5}\) |
| 1 | 6 | \(\frac{1}{6}\) | \(\frac{1}{6}\) |
| 2 | 1 | \(\frac{2}{1}\) | 2 |
| 2 | 2 | \(\frac{2}{2}\) | 1 |
| 2 | 3 | \(\frac{2}{3}\) | \(\frac{2}{3}\) |
| 2 | 4 | \(\frac{2}{4}\) | \(\frac{1}{2}\) |
| 2 | 5 | \(\frac{2}{5}\) | \(\frac{2}{5}\) |
| 2 | 6 | \(\frac{2}{6}\) | \(\frac{1}{3}\) |
| 3 | 1 | \(\frac{3}{1}\) | 3 |
| 3 | 2 | \(\frac{3}{2}\) | \(\frac{3}{2}\) |
| 3 | 3 | \(\frac{3}{3}\) | 1 |
| 3 | 4 | \(\frac{3}{4}\) | \(\frac{3}{4}\) |
| 3 | 5 | \(\frac{3}{5}\) | \(\frac{3}{5}\) |
| 3 | 6 | \(\frac{3}{6}\) | \(\frac{1}{2}\) |
| 4 | 1 | \(\frac{4}{1}\) | 4 |
| 4 | 2 | \(\frac{4}{2}\) | 2 |
| 4 | 3 | \(\frac{4}{3}\) | \(\frac{4}{3}\) |
| 4 | 4 | \(\frac{4}{4}\) | 1 |
| 4 | 5 | \(\frac{4}{5}\) | \(\frac{4}{5}\) |
| 4 | 6 | \(\frac{4}{6}\) | \(\frac{2}{3}\) |
| 5 | 1 | \(\frac{5}{1}\) | 5 |
| 5 | 2 | \(\frac{5}{2}\) | \(\frac{5}{2}\) |
| 5 | 3 | \(\frac{5}{3}\) | \(\frac{5}{3}\) |
| 5 | 4 | \(\frac{5}{4}\) | \(\frac{5}{4}\) |
| 5 | 5 | \(\frac{5}{5}\) | 1 |
| 5 | 6 | \(\frac{5}{6}\) | \(\frac{5}{6}\) |
| 6 | 1 | \(\frac{6}{1}\) | 6 |
| 6 | 2 | \(\frac{6}{2}\) | 3 |
| 6 | 3 | \(\frac{6}{3}\) | 2 |
| 6 | 4 | \(\frac{6}{4}\) | \(\frac{3}{2}\) |
| 6 | 5 | \(\frac{6}{5}\) | \(\frac{6}{5}\) |
| 6 | 6 | \(\frac{6}{6}\) | 1 |
Identifying the Distinct Fractions
Now we list all the simplified fractions from the table above and remove any duplicates to find the unique fractions that form the set S.
- 1
- \(\frac{1}{2}\)
- \(\frac{1}{3}\)
- \(\frac{1}{4}\)
- \(\frac{1}{5}\)
- \(\frac{1}{6}\)
- 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
- \(\frac{5}{2}\)
- \(\frac{5}{3}\)
- \(\frac{5}{4}\)
- \(\frac{5}{6}\)
- 6
- \(\frac{6}{5}\)
Calculating the Cardinality of Set S
By counting the distinct fractions listed above, we find the cardinality of the set S.
There are 23 distinct fractions in the list:
6 fractions from p=1 (1, 1/2, 1/3, 1/4, 1/5, 1/6)
+ 3 new fractions from p=2 (2, 2/3, 2/5)
+ 4 new fractions from p=3 (3, 3/2, 3/4, 3/5)
+ 3 new fractions from p=4 (4, 4/3, 4/5)
+ 5 new fractions from p=5 (5, 5/2, 5/3, 5/4, 5/6)
+ 2 new fractions from p=6 (6, 6/5)
Total distinct fractions = 6 + 3 + 4 + 3 + 5 + 2 = 23.
Thus, the cardinality of set S is 23.
Revision Table: Set Cardinality and Fractions
| Term | Definition | Relevance to Problem |
|---|---|---|
| Set | A collection of distinct objects, called elements. | S is the set whose cardinality we need to find. Its elements are distinct fractions. |
| Cardinality | The number of elements in a set. Denoted by |S|. | We need to find the number of distinct fractions in S. |
| Fraction | A number that represents a part of a whole, written as \(\frac{\text{numerator}}{\text{denominator}}\). In this case, \(\frac{p}{q}\). | The elements of set S are fractions formed using integers from {1, 2, 3, 4, 5, 6}. |
| Distinct Numbers | Numbers that are unique and not equal to each other. | The set S contains only the unique fractions \(\frac{p}{q}\) after simplifying. |
Additional Information: Generating Distinct Fractions
When generating fractions \(\frac{p}{q}\) from a finite set of integers, it's crucial to simplify each fraction to its lowest terms to correctly identify distinct values. For example, \(\frac{2}{4}\) and \(\frac{1}{2}\) represent the same number, so they are not distinct elements in a set like S.
A systematic approach involves:
- Listing all possible combinations of p and q.
- Forming the fraction \(\frac{p}{q}\) for each combination.
- Simplifying each fraction to its simplest form (e.g., by dividing the numerator and denominator by their greatest common divisor).
- Collecting all the simplified fractions.
- Removing any duplicates from the collection to obtain the set of distinct fractions.
- Counting the number of elements in the final set to find the cardinality.
This problem involves a relatively small set of integers {1, 2, 3, 4, 5, 6}, making it feasible to list and check all combinations. For larger sets, more advanced mathematical techniques might be needed, perhaps involving concepts like Euler's totient function if the fractions were restricted to proper fractions in lowest terms within a certain range.
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