All the sets possible

Question

I can eat at USA, France and Italy i can eat at some all or none what are all the sets

Answer 1

First some notation: we define a set to be a collection of distinct objects which we call elements of a set in question. We may explicitly define a set or give conditions on a larger set which restricts us to some subset of a bigger set. If a set contains thing1, thing2 and thing3 then we may denote that set by {thing1, thing2, thing3} i.e in brackets without any repeats (since these are distinct objects) where thing1, thing2 and thing3 are the elements of this set. These may be any objects whatsoever including objects which are sets in their own right, i.e elements of collections of sets since a collection of sets would be a set which itself contains sets as its elements.

Eating at some would be given by the sets {USA, France}, {USA, Italy} and {France, Italy} and note here the order of the objects in these sets does not matter so two sets are the same so long as they contain the same elements, irrespective of order.

Eating at none would be represented by {S such that S is a subset of {USA, France, Italy} which does not contain any elements} = ∅ which denotes the set which only contains itself as a subset and which is itself a subset of every set. In particular, note that any singleton set {thing}, where 'thing' can be anything (so long as it is not a collection that is "too large" in some sense but that's beyond the scope of this problem), is not a subset of ∅ so ∅ does not contain 'thing' and since 'thing' is arbitrary, ∅ doesn't contain anything other than itself and so it is appropriately called the empty set. Notice that ∅ is a unique set since there can only be one set which belong to all sets as a subset and also only contain themselves as a subset since otherwise you would have multiple sets satisfying these properties which would contradict that they only contain themselves. Alternatively the set corresponding to eating at none can be given by {S such that S is a subset of {USA, France, Italy} which does not contain USA, France or Italy.} which again is a set that has no elements and is thus the empty set ∅.

Eating at all corresponds to {S such that S is a subset of {USA, France, Italy} containing USA, France and Italy as elements} = {USA, France, Italy}.