Letter Coding Puzzle - missing edge cases

For the following coding puzzle, I only scored 50% because of 1) performance and 2) completness

My solution:

        HashSet<char> seenLowerCaseLetters = new HashSet<char>();        HashSet<char> seenUpperCaseLetters = new HashSet<char>();        HashSet<char> invalidLetters = new HashSet<char>();        foreach (char s in stringToTest)        {            // Lower Case            if(char.IsLower(s))            {                if (!seenLowerCaseLetters.Contains(s))                {                    seenLowerCaseLetters.Add(s);                    // If Upper case equivalent has been seen, then remove and prevent further addition of this letter                    if (seenUpperCaseLetters.Contains(char.ToUpper(s)))                    {                        invalidLetters.Add(char.ToUpper(s));                        seenUpperCaseLetters.Remove(char.ToUpper(s));                    }                }                // If we've already seen this lower case letter, then try remove from upper case as it would be invalid                else if(seenLowerCaseLetters.Contains(s))                    seenUpperCaseLetters.Remove(char.ToUpper(s));            }            // Lower Case            else if (char.IsUpper(s))            {                // if lower case hasn't been seen already then it can never be a valid letter                if(!seenLowerCaseLetters.Contains(char.ToLower(s)))                    invalidLetters.Add(char.ToUpper(s));                // If lower case has been seen and it's not an invalid letter, add.                if (seenLowerCaseLetters.Contains(char.ToLower(s)) && !invalidLetters.Contains(s))                    seenUpperCaseLetters.Add(s);            }        }        return seenUpperCaseLetters.Count;

Alternative solution:

        Dictionary<char, int> lowerCaseCount = new Dictionary<char, int>();        for(int i = 0; i < stringToTest.Length; i++)        {            char c = stringToTest[i];            if(char.IsLower(c))            {                if (!lowerCaseCount.ContainsKey(c))                    lowerCaseCount.Add(c, 1);                if (lowerCaseCount[c] == 2)                    lowerCaseCount[c] = -1;            }            else            {                char cLower = char.ToLower(stringToTest[i]);                if (!lowerCaseCount.ContainsKey(cLower))                    lowerCaseCount.Add(cLower, -1);                if (lowerCaseCount[cLower] == 1)                    lowerCaseCount[cLower] = 2;            }        }        int count = 0;        foreach(char c in lowerCaseCount.Keys)        {            if (lowerCaseCount[c] == 2)                count++;        }        return count;

Questions:

1. Is the alternative solution much faster?
2. What edge cases have I missed based on the question?

Regarding the performance I can see to a certain point why the alternative is a bit faster but I didn't think HashSets were much slower than arrays.

And I can't see what edge cases are missing.

Thanks