1. Key Concepts and Formulas

• Set Theory: Understanding set notation and how to define elements within a set.
• Number Theory: The concept of the Greatest Common Divisor (GCD) and its property that $\operatorname{gcd}(a, b) = 1$ means $a$ and $b$ are relatively prime (coprime).
• Systematic Enumeration: A method of listing all possible elements that satisfy given conditions to count them accurately.

2. Step-by-Step Solution

Step 1: Understand the Set Definitions and Conditions We are given set $A = \{1, 2, 3, \ldots, 10\}$ and set $B = \left\{\frac{m}{n}: m, n \in A, m<n \text{ and } \operatorname{gcd}(m, n)=1\right\}$. We need to find the number of elements in set $B$, denoted as $n(B)$. The conditions for an element $\frac{m}{n}$ to be in $B$ are:

1. $m \in A$ and $n \in A$: Both $m$ and $n$ must be integers from 1 to 10, inclusive.
2. $m < n$: The numerator must be strictly less than the denominator. This implies that $m$ can range from 1 to 9, and for each $m$, $n$ must be greater than $m$ and at most 10.
3. $\operatorname{gcd}(m, n) = 1$: The numerator and denominator must be relatively prime, meaning the fraction is irreducible.

Step 2: Systematically List Valid Pairs $(m, n)$ We will iterate through all possible values of $m$ from 1 to 9 (since $m<n$ and $n \le 10$, $m$ cannot be 10) and for each $m$, find the possible values of $n$ from the set $\{m+1, m+2, \ldots, 10\}$ such that $\operatorname{gcd}(m, n) = 1$.

• For $m=1$:

  • Possible $n$ values are $\{2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
  • $\operatorname{gcd}(1, n) = 1$ for all $n$.
  • Valid pairs: $(1,2), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10)$.
  • Count: 9.

• For $m=2$:

  • Possible $n$ values are $\{3, 4, 5, 6, 7, 8, 9, 10\}$.
  • We need $\operatorname{gcd}(2, n) = 1$, so $n$ must be odd.
  • Valid $n$: $\{3, 5, 7, 9\}$.
  • Valid pairs: $(2,3), (2,5), (2,7), (2,9)$.
  • Count: 4.

• For $m=3$:

  • Possible $n$ values are $\{4, 5, 6, 7, 8, 9, 10\}$.
  • We need $\operatorname{gcd}(3, n) = 1$, so $n$ must not be a multiple of 3.
  • Valid $n$: $\{4, 5, 7, 8, 10\}$. (Exclude 6, 9).
  • Valid pairs: $(3,4), (3,5), (3,7), (3,8), (3,10)$.
  • Count: 5.

• For $m=4$:

  • Possible $n$ values are $\{5, 6, 7, 8, 9, 10\}$.
  • We need $\operatorname{gcd}(4, n) = 1$. Since $4=2^2$, $n$ must be odd.
  • Valid $n$: $\{5, 7, 9\}$. (Exclude 6, 8, 10).
  • Valid pairs: $(4,5), (4,7), (4,9)$.
  • Count: 3.

• For $m=5$:

  • Possible $n$ values are $\{6, 7, 8, 9, 10\}$.
  • We need $\operatorname{gcd}(5, n) = 1$, so $n$ must not be a multiple of 5.
  • Valid $n$: $\{6, 7, 8, 9\}$. (Exclude 10).
  • Valid pairs: $(5,6), (5,7), (5,8), (5,9)$.
  • Count: 4.

• For $m=6$:

  • Possible $n$ values are $\{7, 8, 9, 10\}$.
  • We need $\operatorname{gcd}(6, n) = 1$. This means $n$ must not be divisible by 2 or 3.
  • Valid $n$: $\{7\}$. (Exclude 8 (divisible by 2), 9 (divisible by 3), 10 (divisible by 2)).
  • Valid pair: $(6,7)$.
  • Count: 1.

• For $m=7$:

  • Possible $n$ values are $\{8, 9, 10\}$.
  • We need $\operatorname{gcd}(7, n) = 1$. Since 7 is prime, $n$ must not be a multiple of 7.
  • Valid $n$: $\{8, 9, 10\}$. (None are multiples of 7).
  • Valid pairs: $(7,8), (7,9), (7,10)$.
  • Count: 3.

• For $m=8$:

  • Possible $n$ values are $\{9, 10\}$.
  • We need $\operatorname{gcd}(8, n) = 1$. Since $8=2^3$, $n$ must be odd.
  • Valid $n$: $\{9\}$. (Exclude 10 (even)).
  • Valid pair: $(8,9)$.
  • Count: 1.

• For $m=9$:

  • Possible $n$ value is $\{10\}$.
  • We need $\operatorname{gcd}(9, n) = 1$. Since $9=3^2$, $n$ must not be a multiple of 3.
  • Valid $n$: $\{10\}$. (10 is not a multiple of 3).
  • Valid pair: $(9,10)$.
  • Count: 1.

Step 3: Sum the Counts to Find $n(B)$ The total number of elements in set $B$ is the sum of the counts from each case of $m$: $n(B) = 9 + 4 + 5 + 3 + 4 + 1 + 3 + 1 + 1 = 31$

3. Common Mistakes & Tips

• Forgetting GCD Condition: Failing to check $\operatorname{gcd}(m, n)=1$ will lead to including reducible fractions like $\frac{2}{4}$ or $\frac{3}{6}$, overcounting the elements.
• Incorrect Range for $n$: Ensure $n$ is always strictly greater than $m$ and not exceeding 10. For example, when $m=9$, $n$ can only be 10.
• Systematic Checking: For composite $m$, remember that $\operatorname{gcd}(m, n)=1$ implies $n$ is coprime to all prime factors of $m$. For instance, $\operatorname{gcd}(6, n)=1$ means $n$ is not divisible by 2 and not divisible by 3.

4. Summary We identified the conditions for elements in set $B$: $m, n$ are from $\{1, \ldots, 10\}$, $m < n$, and $\operatorname{gcd}(m, n) = 1$. By systematically iterating through all possible values of $m$ from 1 to 9 and finding the corresponding valid values of $n$ that satisfy $m < n \le 10$ and $\operatorname{gcd}(m, n) = 1$, we counted the total number of such pairs. Summing the counts for each $m$ gives the total number of elements in set $B$.

5. Final Answer The total number of elements in set $B$ is $31$. This corresponds to option (A). The final answer is $\boxed{\text{31}}$.