Set union

Published

2024-02-02

Set union is defined thus:

Let A and B be sets. The union of A and B, written A \cup B, is the set of all elements in A and all elements in B. In set builder notation:

A \cup B = \{x \: | \: x \in A \text{ or } x \in B \}.

The empty set is the identity element for this operation, so

A \cup \emptyset = A.

This is because the empty set contains no elements, and thus can add nothing to the union.

We can take the union of more than two sets, and we notate this with a big cup. For example,

A_1 \cup A_2 \cup \ldots \cup A_n = \bigcup\limits_{i = 1}^n A_i

for an indexed family of sets, or sometimes

\bigcup\limits_{A \in R} A

for some set of sets, R.