If \(S=\{1,2,3\}\) and \(A=\mathcal{P}(S)\), then a relation \(R\) on \(A\) is defined by \[R=\{(X,Y):X,Y\in A \text{ and } X\cup =S\}.
\] How many diagonal pairs must be added to make \(R\) reflexive?

Correct answer: 7

Step 1: \(\mathcal{P}(S)\) has (8) elements, so (8) diagonal pairs are needed. Step 2: On the diagonal, (X\cup X=X), which equals (S) only when (X=S). Step 3: One diagonal pair is already present, so (7) must be added.