# Right quotient

If [itex]L_1[itex] and [itex]L_2[itex] are formal languages, then the right quotient of [itex]L_1[itex] with [itex]L_2[itex] is the language consisting of strings w such that wx is in [itex]L_1[itex] for some string x in [itex]L_2[itex]. In symbols, we write:

[itex]L_1 / L_2 = \{w \ | \ \ \exists x \in L_2 \ \ : \ \ wx \in L_1\}[itex]

Some common closure properties of quotient include:

There is a related notion of left quotient, which keeps the postfixes of [itex]L_1[itex] without the prefixes in [itex]L_2[itex]. Sometimes, though, "right quotient" is written simply as "quotient". The above closure properties hold for both left and right quotients.

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