Calculating the number of coprime subsets of a set involves a combination of combinatorial counting and an understanding of the properties of numbers. Specifically, a subset of a set is considered coprime if the greatest common divisor (gcd) of all its elements is 1. This article will explain how to calculate the number of coprime subsets for the set ${1, 2, 3, \ldots, n}$.

Steps to Calculate Number of Coprime Subsets

Understanding Coprimality

Two integers are coprime if their gcd is 1. For a subset to be coprime, all pairs of numbers within the subset should be coprime. That is, no subset can have a gcd greater than 1.

Inclusion-Exclusion Principle

To determine the number of coprime subsets, we use the principle of inclusion-exclusion, which allows us to count the number of elements with overlapping properties.

Euler's Totient Function

Euler's Totient Function $\phi(n)$ is crucial for this computation. It provides the number of integers up to $n$ that are coprime with $n$. However, to find coprime subsets, one must consider subsets of all sizes, which complicates the direct use of $\phi(n)$.

Coprime Subset Count Formula

We can derive the number of coprime subsets using a combination of inclusion-exclusion and properties of Euler's Totient Function. Let $S = {1, 2, \ldots, n}$.

1. Initial Subset Count:
   The total number of subsets of $S$ is $2^n$ (including the empty set).
2. Exclusion of Non-Coprime Subsets:
   A subset is not coprime if all its elements share a common factor greater than 1. By considering all primes $p \leq n$, we exclude subsets that are not coprime:

$$\text{Exclude with: } \sum\_{p \leq n} (2^{n/\text{p}} - 1)$$

1. Inclusion-Exclusion for Non-Coprime Subsets:
   Use the inclusion-exclusion principle to properly account for overlaps:

$$\text{After inclusion-exclusion,} = 2^n - \sum\_{p \leq n} 2^{n/p} + \sum\_{p \< q \leq n} 2^{n/(pq)} - \cdots$$

1. Valid Coprime Subsets:
   Finally, the number of coprime subsets is obtained by subtracting non-coprime subsets derived from the inclusion-exclusion principle from the total subsets $2^n$.

$$|{\text{coprime subsets}}| = 2^n - \sum\_{R} 2^{\left \lceil \frac{n}{\text{lcm}(r\_1, r\_2, \ldots, r\_k)} \right \rceil}$$

where $R$ ranges over sets of primes dividing some subset of $1, 2, \ldots, n$.

Example Calculation

Consider $n = 4$. The set is ${1, 2, 3, 4}$.

1. Total subsets: $2^4 = 16$.
2. Identify non-coprime subsets: • $2^{4/2} = 4$ subsets divisible by $2$: ${2}, {4}, {2, 4}$ • $2^{4/3} = 2$ subset divisible by $3$: ${3}$
3. Coprime subsets via exclusion: • Exclude subsets: $4 + 2 - 1 = 5$ • Coprime subsets: $16 - 5 = 11$

Hence, there are 11 coprime subsets for the set ${1, 2, 3, 4}$.

Key Points Summary Table

Additional Considerations

• Performance: Computing coprime subsets can get computation-intensive for large $n$. Efficient use of mathematical libraries and optimization techniques like memoization can be beneficial. • Application: This method is relevant for understanding combinatorial structures, coding theory, and network topology where independent interactions are modeled.

By understanding these concepts and using the formula described, you can reliably calculate the number of coprime subsets of any set ${1, 2, 3, \ldots, n}$.