Why is the empty set a subset of all sets?

Answer 1

A subset of a given set is simply any set, all of whose elements are contained in the other. Since the empty set has no elements, all of its elements are in any other set! It sounds weird, but that's the way logic works. To put it another way, a set A is NOT a subset of B if there is some element x of A that is not in B. Since the empty set has no elements that are not in your given set, we can't say it is NOT a subset. That means that it is. To select a subset, we must look at each member of the set and decide whether to keep it. If we say "yes" to every member, we have the set itself; if we say "no" to all of them, we have the empty set. We could choose to exclude these from the definition of subset, but it makes a lot of things easier if we include them. That way there are no special cases to deal with when we state theorems.