The Difference Between {} and {{}}

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Abstract

To the mathematically naïve, these two entities can cause some perplexity. In this knol we show the significant differences between these two mathematical objects.

In this context {} means the empty set, and {{}} means the set containing only the empty set. They are not the same thing, because {} contains no members, while {{}} contains the empty set as a member. To look at it another way, consider the collection of subsets of the set {a,b,c}. These form the collection {{},{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c}}. Now remove all nonempty sets from this collection. What remains is {{}}. Finally, pick any collection S of subsets of {a,b,c}, for example we might take S={{a,b},{a,c},{b,c},{a,b,c}}. Then the intersection of S with {{}} equals {}, since {} was not a member of S. If instead we had chosen for example S={{},{a,b}}, then the intersection would have been {{}}.

In fact, one definition of {{}} might be the intersection of two collections with only the empty set as a common member.