In each square of an $m \times n$ rectangular board there is a nought or a cross. Let $f(m,n)$ be the number of such arrangements that contain a row or a column consisting of noughts only. Let $g(m,n)$ be the number of arrangements that contain a row consisting of noughts only, or a column consisting of crosses only. Which of the numbers $f(m,n)$ and $g(m,n)$ is larger?
1. **Define the problem and notation:**
   - We have an \(m \times n\) rectangular board where each square contains either a nought or a cross.
   - Let \(f(m,n)\) be the number of arrangements that contain at least one row or one column consisting entirely of noughts.
   - Let \(g(m,n)\) be the number of arrangements that contain at least one row consisting entirely of noughts or at least one column consisting entirely of crosses.

2. **Define \(R(m,n)\) and \(L(m,n)\):**
   - \(R(m,n)\) is the number of arrangements that contain at least one row consisting entirely of noughts.
   - \(L(m,n)\) is the number of arrangements that contain at least one column consisting entirely of noughts.

3. **Inclusion-Exclusion Principle for \(f(m,n)\):**
   - By the Inclusion-Exclusion Principle, the number of arrangements that contain at least one row or one column consisting entirely of noughts is:
     \[
     f(m,n) = R(m,n) + L(m,n) - \text{(number of arrangements that contain both a row and a column consisting entirely of noughts)}
     \]

4. **Inclusion-Exclusion Principle for \(g(m,n)\):**
   - Similarly, the number of arrangements that contain at least one row consisting entirely of noughts or at least one column consisting entirely of crosses is:
     \[
     g(m,n) = R(m,n) + L(m,n) - \text{(number of arrangements that contain both a row of noughts and a column of crosses)}
     \]

5. **Comparing \(f(m,n)\) and \(g(m,n)\):**
   - Note that the number of arrangements that contain both a row and a column consisting entirely of noughts is at least one (the board without crosses).
   - The number of arrangements that contain both a row of noughts and a column of crosses is less than or equal to the number of arrangements that contain both a row and a column consisting entirely of noughts.
   - Therefore, the subtraction term in \(f(m,n)\) is larger than or equal to the subtraction term in \(g(m,n)\).

6. **Conclusion:**
   - Since the subtraction term in \(f(m,n)\) is larger than or equal to the subtraction term in \(g(m,n)\), it follows that:
     \[
     f(m,n) < g(m,n)
     \]

The final answer is \( \boxed{ f(m,n) < g(m,n) } \)