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:

\(\ds S_1\)

\(=\)

\(\ds \set {a, b}\)

\(\ds S_2\)

\(=\)

\(\ds \set {b, c}\)

\(\ds S_3\)

\(=\)

\(\ds \set {a, c}\)

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:

\(\ds a\)

\(\notin\)

\(\ds S_3\)

as $S_3 = \set {a, c}$

\(\ds b\)

\(\notin\)

\(\ds S_2\)

as $S_2 = \set {b, c}$

\(\ds c\)

\(\notin\)

\(\ds S_1\)

as $S_1 = \set {a, b}$

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

That is:

$\ds \bigcap_{i \mathop \in I} S_i = \set {x: \forall i \in I: x \in S_i} = \O$

That is:

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

However, note that:

\(\ds S_1 \cap S_2\)

\(=\)

\(\ds \set b\)

\(\ds \ne \O\)

\(\ds S_2 \cap S_3\)

\(=\)

\(\ds \set c\)

\(\ds \ne \O\)

\(\ds S_1 \cap S_3\)

\(=\)

\(\ds \set a\)

\(\ds \ne \O\)

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.

$\blacksquare$