Union of Blocks is Set of Points

Theorem
Let $\struct {X, \mathcal B}$ be a pairwise balanced design.

That is, let $\struct {X, \mathcal B}$ be a design, with $\size X \ge 2$, and the number of occurrences of each pair of distinct points in $\mathcal B$ be $\lambda$ for some $\lambda > 0$ constant.

Then the set union of all the subset elements in $\mathcal B$ is precisely $X$.

Lemma
Let $\struct {X, \mathcal B}$ be a balanced incomplete block design.

Then the set union of all the subset elements in $\mathcal B$ is precisely $X$.

Proof
Let $X = \set {x_1, x_2, \ldots, x_v}$.

Let $\mathcal B = \multiset {y_1, y_2,\ldots, y_b}$, where the notation denotes a multiset.

Let $Y = \displaystyle \bigcup_{i \mathop = 1}^b y_i$.

We shall show that $Y \subseteq X$ and $X \subseteq Y$.

$Y \subseteq X$:
By definition, $\mathcal B$ is a multiset of subsets of $X$.

This means that each $y_i \in Y$ is a subset of $X$ for $i \in \closedint 1 b$.

By Union of Subsets is Subset, $Y$ itself is a subset of $\mathcal B$.

$X \subseteq Y$:
Let $x_i \in X$ be arbitrary.

Choose any $x_j \in X$ such that $i \ne j$.

Such an $x_j$ necessarily exists because by hypothesis $\card X \ge 2$.

Then $\set {x_i, x_j} \subseteq Y$ by the definition of blocks, as $\lambda > 0$ by hypothesis.

But by the definition of subset, this implies that $x_i$ is an element in $Y$.

Hence the result, as $x_i$ was arbitrary.

Proof of Lemma
By the definition of balanced incomplete block design, $\struct {X, \mathcal B}$ is itself a type of pairwise balanced design.

Thus the lemma follows from the main result.