Definition:Disjoint Union

Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

Disjoint Union may refer to:

Set Theory

Let $\family {S_i}_{i \mathop \in I}$ be an $I$-indexed family of sets.

The disjoint union of $\family {S_i}_{i \mathop \in I}$ is defined as the set:

$\ds \bigsqcup_{i \mathop \in I} S_i = \bigcup_{i \mathop \in I} \set {\tuple {x, i}: x \in S_i}$

where $\bigcup$ denotes union.

Each of the sets $S_i$ is canonically embedded in the disjoint union as the set:

${S_i}^* = \set {\tuple {x, i}: x \in S_i}$

Symmetric Difference

The symmetric difference between two sets is also known as their:

disjoint union

Boolean sum.

However, both terms have different or more specialized meanings on $\mathsf{Pr} \infty \mathsf{fWiki}$, so will not be used here.

Another term seen occasionally is symmetric sum.

Some sources are pedantically explicit and use the term symmetric difference set.

Probability Theory

Let $\CC$ be a collection of pairwise disjoint sets.

That is, for all sets $A, B \in \CC: A \ne B \implies A \cap B = \O$.

Then the union of all sets in $\CC$ is called their disjoint union:

$\ds \bigsqcup_{A \mathop \in \CC} A \equiv \bigcup_{A \mathop \in \CC} A$

That is, in this context the term disjoint union means union of sets which are pairwise disjoint.

Also see

• Definition:Disjoint
• Definition:Union