is a null set the same as {} Ø vs set {}:
Ø has no elements
whereas {} has the null set as an element; that is, say you are making power set of {}: Ø would be an element of the power set or in symbols, P({}): {Ø}
right? I feel these distinctions might come into play in certain problems I come across; they are confusing and I am looking for clarification
I ask because I do not think Ø = {}
 A: When one doubts whether a given set is equal to "another" set, one can use the axiom of extentionnality ( the definition of set equality) according to which : 
(set A = set B)  IFF ( for all x,  x belongs to A <--> x belongs to B) . 
But using this axiom requires, in certain  cases, some  familiarity with the truth table of the " if...then" and  of the " <--> " operator. One has to know that, when a set S is empty, the sentence " x belongs to S" being false, any conditional  statement having the sentence " x belongs to S" as antecedent is " automatically" true ( "vacuously true" as some say pleasantly).  And notice that a biconditionnal ( " <-->" ) is simply the conjunction of two " if ...then" statements. 
Here if you plug   the set Ø and the set    {  } in the formula of the axiom of extentionnality, you will see you will get a statement that is vacuously true ( in both directions). So your "two" sets pass the test of equality: they are, in fact, exactly the same set. 
If you are not acquainted  with the " if... then" or " iff" operator, rephrase the principle of extensionnality, and say that a set A and a set B are equal just in case : 
(1) there is no object that belongs to A and that does not belong to B 
AND 
(2) there is no object that belongs to B and that does not belong to A. 
So can you see an object that belongs to Ø and does not belong to {  } ? and can you see an object that belongs to {  } but does not belong to Ø ? 
If you answered "no" to both questions, you know that : Ø = {  }  ( or, if you prefer, that : {  } = Ø ). 
A: Let us make it clear by defining each set.
$1)$  The empty set $\phi$ is a set with no element. We use it for example to denote the intersection of tow disjoint sets.
$2)$ $\{\}$ is another way to show the empty set so it does not have any element.
$3)$ $\{\phi\}$ is a set which has one element and that one element is the empty set. 
$4)$ The power set of the empty set $P(\phi)$ is the set of subsets of the empty set. That is $\{\phi\}$ the same as the set in $3)$ 
