How many Subsets does a Null set has? 2 or 1 
How can this Question's answer be 2? If it was the number of elements for the set {Φ} I would have gone with 2, But how is the Answer this question be 2?
 A: (1) The null set is a subset of every set; so the null set certainly has at least one subset, namely itself.
(2) Now suppose that S is a second subset of the null set. in that case, we have :
$(a)\Large S\subseteq \emptyset$
and
$(b)\Large S\neq \emptyset $
(3) The extensionnality axiom tells us that two sets must differ by at least one element in order to be different. So $(b)$ implies that : either $\emptyset$ has an element that does not belong to $S$ , or $S$ has an element that does not belong to $\emptyset$. But the first option is impossible ( by the definition of the empty set), so only the second remains.
(4) The second option says : there is at least some object $x$ such that $\Large  x\in S$ and $\Large x\notin \emptyset$, meaning ( by &-elimination) that there is at least some object $x$ such that $\Large x\in S$.  But , by $(2.a)$ above , $\Large S\subseteq \emptyset$, which means that all elements of $S$ are also elements of $\emptyset$. This holds in particular for object $x$. So, the second option leads to : $\Large x\in \emptyset$ , another impossibility.
(5) Conclusion : the empty set cannot have any subset in addition to itself; the number of its subsets is $1$.
