Suppose \(S\) were the closure of \(R\) with respect to this property. Since \(R\) does not have an odd number of elements, \(S \ne R\), so \(S\) must be a proper superset of R.
Clearly S cannot have more than 5 elements, for if it did, then any subset of \(S\) consisting of \(R\) and one element of \({\rm{S}}\) - \({\rm{R}}\) would be a proper subset of \({\rm{S}}\) with the property; this would violate the requirement that \({\rm{S}}\) be a subset of every superset of .\(R\). with the property.
Thus \({\rm{S}}\) must have exactly 5 elements.
Let T be another superset of \(R\) with 5 elements (there are \(9 - 4 = 5\) such sets in all).
Thus T has the property, but \(S\) is not a subset of \(T\).
This contradicts the definition.
Therefore our original assumption was faulty, and the closure does not exist.