Union of Blocks is Set of Points/Corollary
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Corollary to Union of Blocks is Set of Points
Let $\struct {X, \BB}$ be a balanced incomplete block design.
Then the set union of all the subset elements in $\BB$ is precisely $X$.
Proof
By the definition of balanced incomplete block design, $\struct {X, \BB}$ is itself a type of pairwise balanced design.
Thus the corollary follows from Union of Blocks is Set of Points.
$\blacksquare$