# Are all finite languages context-free?

As far as I know, finite languages have a finite number of strings or words, while context-free languages are generated by context-free grammars. I don't know which aspect should I know that they are correlated with. All I know is that finite languages are regular, and regular languages are context-free. How should I know this?

• "All I know is that finite languages are regular, and regular languages are context-free." This immediately implies that all finite languages are context-free. Jan 12, 2021 at 11:09

The language consisting of the words $$w_1,w_2,\ldots,w_n$$ is generated by the context-free grammar $$S \to w_1 \mid w_2 \mid \cdots \mid w_n.$$

• So, are all finite languages context-free sir? or just some? Jan 12, 2021 at 5:49
• @JorelleTuyor What this answer is telling you: Every finite language is regular, because it can be created with a regular grammar like above, where the words are simply exhaustively listed. So yes, every finite language is regular (and thus also context free). Jan 12, 2021 at 7:00
• @kutschkem Thank you for clarifying sir. God bless. Jan 12, 2021 at 7:08

Let

• $$F$$ be the set of finite languages
• $$R$$ be the set of regular languages
• $$C$$ be the set of context-free languages.

The statement "All finite languages are regular" can be rewritten $$F \subseteq R$$. Similarly, "All regular languages are context-free" can be rewritten $$R \subseteq C$$. Both being true, we can take $$F \subseteq R \land R \subseteq C$$ to be true as well.

Set inclusion is known to be a transitive relation, that is $$F \subseteq R \land R \subseteq C \rightarrow F \subseteq C$$. Thus, by modus ponens, we can conclude that $$F \subseteq C$$ is true, or that all finite languages are context-free.

• Very Well said Sir. Thanks.❤ Jan 12, 2021 at 15:21