Deterministic Context-free Languages NOT Closed Under Union or Intersection

Here we give a proof that deterministic context-free languages (DCFLs) are NOT closed under union. We use the prototypical not-DCFL, the complement of {a^n b^n c^n : n at least 0}, and show that it is the union of regular languages and three DCFLs. Since DCFLs are closed under union with regular languages, then union-ing the three DCFLs must result in a language that isn't a DCFL. We also get that they're not closed under intersection for free.

A deterministic context-free language is the language of a deterministic pushdown automaton (DPDA). The essential idea is that a DPDA is equivalent to a PDA, except that there can only be one "choice" of transition from any state on any input, with anything on the stack. See https://www.youtube.com/watch?v=GzR5FiiIogY&ab_channel=EasyTheory for more details.

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