Disjoint Family of Sets/Examples/3 Arbitrary Sets

Examples of Disjoint Families of Sets
Let $I = \set {1, 2, 3}$ be an indexing set.

Let:

Then the family of sets $\family {S_i}_{i \mathop \in I}$ is disjoint, but not pairwise disjoint.

Proof
Let $S = {a, b, c}$, and so:
 * $S_1, S_2, S_3 \subseteq S$

We have that:

Thus there is no element of $S$ which is also an element of all of $S_1$, $S_2$ and $S_3$.

That is:
 * $\displaystyle \bigcap_{i \mathop \in I} S_i = \set {x: \forall i \in I: x \in S_i} = \O$

That is:
 * $\family {S_i}_{i \mathop \in I}$ is disjoint.

However, note that:

Thus it is noted that while $\family {S_i}_{i \mathop \in I}$ is disjoint, it is not the case that $\family {S_i}_{i \mathop \in I}$ is pairwise disjoint.