Quotient of a formal language

In mathematics and computer science, the right quotient (or simply quotient) of a language ${\displaystyle L_{1}}$ with respect to language ${\displaystyle L_{2}}$ is the language consisting of strings w such that wx is in ${\displaystyle L_{1}}$ for some string x in ${\displaystyle L_{2}}$.[1] Formally:

$${\displaystyle L_{1}/L_{2}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ wx\in L_{1}\}}$$

In other words, we take all the strings in ${\displaystyle L_{1}}$ that have a suffix in ${\displaystyle L_{2}}$, and remove this suffix.

Similarly, the left quotient of ${\displaystyle L_{1}}$ with respect to ${\displaystyle L_{2}}$ is the language consisting of strings w such that xw is in ${\displaystyle L_{1}}$ for some string x in ${\displaystyle L_{2}}$. Formally:

$${\displaystyle L_{2}\backslash L_{1}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ xw\in L_{1}\}}$$

In other words, we take all the strings in ${\displaystyle L_{1}}$ that have a prefix in ${\displaystyle L_{2}}$, and remove this prefix.

Note that the operands of ${\displaystyle \backslash }$ are in reverse order: the first operand is ${\displaystyle L_{2}}$ and ${\displaystyle L_{1}}$ is second.