Lesson 3

For any set \(A\), the empty set is a subset of \(A\): \(\varnothing \subseteq A\). But a set can also contain the empty set. For example, \(\varnothing\), \(\{\varnothing\}\), \(\{\{\varnothing\}\}\), \(\{\varnothing, \{\varnothing\}\}\) are four different sets:

• \(\varnothing=\{\}\) is the empty set (it has size zero);


• \(\{\varnothing\}\) is the set whose only element is the empty set (it has size one);


• \(\{\{\varnothing\}\}\) is the set whose only element is the set containing the empty set (it has size one);


• \(\{\varnothing, \{\varnothing\}\}\) is the set containing the empty set and the set consisting of the empty set (it has size two).