Empty set
If a set does not contain any elements, then the set is the unique Empty Set. The Empty Set is the most basic set and has a reserved symbol, \(\emptyset.\) The value \(0\) is to the special set of all real numbers, \(R\), as the Empty Set is to any set S. By this we meant that the contents of the Empty Set can be adjoined to the list of elements in any set S without changing the set S just as \(0\) can be added to any real number without chaning its value.
The Empty Set is an empty structure that is contained in every set. Since the Empty Set is contained in every subset \(S,\) we say that the \(\emptyset \subseteq S\) for all sets \(S.\) For now, imagine the term “subset” to mean “contained inside of.” The Empty Set is contained inside of every possible set. Using the symbol \(\subset\) to indicate containment, we can symbolically write that the Empty Set is contained in any set \(S\) as follows:
The Empty set (\(\emptyset\)) is the smallest set and is contained inside every other set in the Universal set.
Let \(S\) be any set, including the Empty Set.
\(\emptyset \subset S.\)
To illustrate consider the the Venn diagram with our Universal set \(U\) as the gray rectangle containing two sets \(O\) and \(B.\) The Empty set is inside all of these sets.