Wahab,

Yes, a set is a collection of well defined objects but there is no restriction on the number of objects in the set. You may be able to describe the set in terms of well defined objects but it just happens that no object satisfies the requirements. The set of all positive integers that are both even and odd is such a set. The terms positive, integer, even and odd are all well defined concepts. Even integers are divisible by 2 and odd integers are not divisible by 2. Although the defining property "all positive integers that are both even and odd" is a perfectly well formed mathematical statement there is no positive integer that satisies this requirement. Hence the set of all positive integers that are both even and odd is empty.

It is very useful to have the empty set as a valid mathematical object. In particular one of the operations we want to perform with sets is intersection. The intersection of two sets is the set of all objects that are in both sets. Thus the intersection of the set {a, b, c, d} and {c, d, e} is the set {c, d}. I want the intersection of two sets to be a set, But what happens with the intersection of the set {a, b, c, d} and the set {p, q, r, s}? There are no letters that are in both sets so the intersection is empty. If I want the intersection of two sets to always be a set then this empty collection must be a set.

I hope this helps,
Harley