A very good question, and one I suspect people don't ask often enough! Unfortunately, the answer is overly complicated (it is based on a set of axioms which are not nice at all, called `ZFC'). However, wikipedia seems to cover the topic well, so I would recommend reading that page then trying your question. If you still can't crack it, ask it again!

Anyway, a basic notion of sets is a collection of objects, called elements. These elements are normally numbers (because this is maths). To show that the collection of elements is a set we enclose it in curly brackets, for example is a set. An exampls of an element of would be 3, and 4 would be another one. 6, however, is not an element.

Sets cannot have more than one copy of an element. So, for example, .

Now, there are a few operations defined on sets. The main ones are union and intersection.

then cancel any multiple elements. So, for example, .

Intersection gives you the subset of elements contained in both of the sets. So, for example, as the elements 1 and 4 are the only elements in both of the sets.

Does that make sense?