VERY simple C# Set implementation
I'm going to assume that you already know about HashSet and ISet in the following review and are just missing the reinventing-the-wheel tag.

ICollection

You've got Add, Remove, Clear, Count, and Contains. If you add a CopyTo(T[] array, int arrayIndex) and bool IsReadOnly { get { return false; } } to your implementation then you can implement ICollection.

Why Dictionary?

Perhaps you'd be better suited with a HashSet&lt;T&gt; implementation? Correct me if I'm wrong but it doesn't appear you're actually using the boolean value of the dictionary, so you'd be better suited with a Hashset&lt;T&gt;.

Generics

You could create a generic implementation Set&lt;T&gt; fairly easily. And if you wanted to still allow your existing functionality then you'd use Set&lt;object&gt;.

Set Operations

Perhaps consider having your operator functions return the new set from their actions?

For example, 

public Set UnionWith(Set input)
{
    var unionedSet = new Set(_items); // See suggestion below.
    foreach (var item in input.Where(x =&gt; _items.ContainsKey(item)))
    {
        unionedSet.Add(item, true);
    }

    return unionedSet;
}


Constructor

Adding the ability to give IEnumerable or ICollection of items in a constructor could be really helpful, especially if you have your set operations return new sets as suggested above.