is a null set the same as {}

Question

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 = {}

Answer 1

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 ).

Answer 2

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 $3)$