what's the difference between the empty set and {empty set}???

Answer 1

The empty set has no elements.
{the empty set} is a set with one element: the empty set.

The distinction is very important in set theory and arithmetic. Author Murray Eisenberg described it as important as distinguishing between a lion and a cage holding the lion!

In the foundations of arithmetic, one can build the natural (counting) numbers by defining the number 0 to be the empty set, the number 1 to be {empty set), the number 2 to be

{empty set, {empty set}},

and so on. Thus 0 is a set with no elements, 1 has 1 element, 2 has 2 elements, and so on. There are axioms in set theory which assure that all of these constructions are well-formed, and are sets.

Answer 2

An empty set does not contain anything; {empty set} is a set containing something, which is an empty set.

Answer 3

The empty set does not have anything in it. Its like a bag with nothing in it. [empty set] is like a bag with another bag in it. When you look into the bag inside the other bag you see nothing. [empty set] has one item in it, namely the empty set. Empty set does not have anything in it.

Answer 4

i guess empty set has nothing in it
but {empty set} is a set that has one element, therefore has cardinality one
further {empty set,{empty set}} has cardinality two and so on.
ok you got the idea but let me write the cardinality 3, it is funny:
{empty set, {empty set},{empty set,{empty set}}}
I say cardinality 3 because I think the similarity goes further
and you can say that
1={empty set}
2={empty set,{empty set}}
3={empty set, {empty set},{empty set,{empty set}}},
defining(merging) the number with its cardinality.

Answer 5

None that I know of.

Sets are placed in { } usually, but both should mean the exact same thing.