the union operation

Sometimes a pattern allows more than one possibility, like a form that accepts either 'Mr' or 'Ms'. The union operation in a regular expression is the way to say 'either this pattern or that one'. If you can describe two families of strings separately, union lets you accept anything from either family.

If R and S are regular expressions, the union R+S (often written R|S in programming) denotes the set union of their languages: a string matches R+S exactly when it matches R or it matches S (or both). In set notation, L(R+S) = L(R) ∪ L(S), where ∪ is set union. For example, over {a, b}, the expression a+b denotes { a, b }, and (ab)+(ba) denotes { ab, ba }. Union is the regular-expression operator that corresponds directly to the everyday word 'or'.

Union is commutative and associative (the order and grouping of alternatives do not matter) and it is idempotent: R+R denotes the same language as R, because a set union with itself adds nothing. The empty set ∅ is its identity element: R+∅ denotes the same language as R, since unioning in 'no strings' changes nothing. These facts are part of the algebra of regular expressions and let you simplify big expressions.