CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Regular languages are closed under the Boolean operations over a fixed alphabet

Statement

Fix an alphabet $Σ$ . If $L, K \subseteq Σ^{*}$ are regular, then every Boolean combination of $L$ and $K$ inside the same ambient $Σ^{*}$ is also regular.

Facts & Assumptions

Given: Regular languages $L, K \subseteq Σ^{*}$ .

[L1]

Regular languages are exactly the languages recognized by DFA's, by Regular languages.

[L2]

The product construction gives DFA's for union and intersection, by The product construction gives DFA's for union and intersection.

[L3]

Complementing the accepting states complements the recognized language, by Complementing the accepting states complements the recognized language.

Proof

technique · direct

1.1

By [L1], choose DFA's recognizing $L$ and $K$ . Then [L2] gives DFA's recognizing $L \cup K$ and $L \cap K$ , and [L3] gives a DFA recognizing $Σ^{*} ∖ L$ .

givenL1L2L3

2.1

The remaining Boolean operations are obtained from union, intersection, and complement by ordinary set identities inside the fixed ambient set $Σ^{*}$ . For instance, $L ∖ K = L \cap (Σ^{*} ∖ K)$ and the symmetric difference is $(L ∖ K) \cup (K ∖ L)$ .

algebrastep 1.1

3.1

Therefore every Boolean combination of $L$ and $K$ over the fixed alphabet $Σ$ is regular.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, Introduction to the Theory of Computing, Lecture 4 (standard reference, not scraped)