quotient of languages
Let ${L}_{1},{L}_{2}$ be two languages^{} over some alphabet $\mathrm{\Sigma}$. The quotient^{} of ${L}_{1}$ by ${L}_{2}$ is defined to be the set
$${L}_{1}/{L}_{2}:=\{u\in {\mathrm{\Sigma}}^{*}\mid uv\in {L}_{1}\text{for some}v\in {L}_{2}\}.$$ 
${L}_{1}/{L}_{2}$ is sometimes written ${L}_{1}{L}_{2}^{1}$. The quotient so defined is also called the right quotient, for one can similarly define the left quotient of ${L}_{1}$ by ${L}_{2}$:
$${L}_{1}\backslash {L}_{2}:=\{u\in {\mathrm{\Sigma}}^{*}\mid vu\in {L}_{1}\text{for some}v\in {L}_{2}\}.$$ 
${L}_{1}\backslash {L}_{2}$ is sometimes written ${L}_{2}^{1}{L}_{1}$.
Below are some examples of quotients:

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If ${L}_{1}=\{{a}^{n}{b}^{n}{c}^{n}\mid n\ge 0\}$ and ${L}_{2}={\{b,c\}}^{*}$, then

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${L}_{1}/{L}_{2}=\{{a}^{m}{b}^{n}\mid m\ge n\ge 0\}$

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${L}_{2}/{L}_{1}={L}_{2}$

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${L}_{1}\backslash {L}_{2}=\{\lambda \}$, the singleton consisting the empty word

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${L}_{2}\backslash {L}_{1}={L}_{2}$

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for any language $L$ over $\mathrm{\Sigma}$:

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$L/{\mathrm{\Sigma}}^{*}$ is the language of all prefixes of words of $L$

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${\mathrm{\Sigma}}^{*}/L={\mathrm{\Sigma}}^{*}$

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$L\backslash {\mathrm{\Sigma}}^{*}$ is the language of all suffixes of words of $L$

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${\mathrm{\Sigma}}^{*}\backslash L={\mathrm{\Sigma}}^{*}$

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$\lambda \in L/L\cap L\backslash L$, and if $\lambda \in L$, then $L\subseteq L/L\cap L\backslash L$.
Here are some basic properties of quotients:

1.
${L}_{1}\subseteq ({L}_{1}/{L}_{2}){L}_{2}\cap {L}_{2}({L}_{1}\backslash {L}_{2})$.

2.
$({L}_{1}/{L}_{2}){L}_{2}\subseteq ({L}_{1}{L}_{2})/{L}_{2}$, and ${L}_{2}({L}_{1}\backslash {L}_{2})\subseteq ({L}_{2}{L}_{1})\backslash {L}_{2}$.

3.
right and left quotients are related via reversal:
${({L}_{1}\backslash {L}_{2})}^{R}$ $=$ $\mathrm{\{}{u}^{R}\mid vu\in {L}_{1}\text{for some}v\in {L}_{2}\}$ $=$ $\mathrm{\{}{u}^{R}\mid {(vu)}^{R}\in {L}_{1}^{R}\text{for some}{v}^{R}\in {L}_{2}^{R}\}$ $=$ $\mathrm{\{}{u}^{R}\mid {u}^{R}{v}^{R}\in {L}_{1}^{R}\text{for some}{v}^{R}\in {L}_{2}^{R}\}$ $=$ ${L}_{1}^{R}/{L}_{2}^{R}.$
A family $\mathcal{F}$ of languages is said to be closed under quotient by a language $L$ if for every language $M\in \mathcal{F}$, $M/L\in \mathcal{F}$. Furthermore, $\mathcal{F}$ is said to be closed under quotient if $M/L\in \mathcal{F}$ for any $M,L\in \mathcal{F}$. Closure under quotient is also termed closure under right quotient. Closure under left quotient is similarly defined.
It can be shown that the families of regular, contextfree, and type0 languages are closed under quotient (both left and right) by a regular language. The family of contextsensitive languages does not have this closure property.
Since all of the families mentioned above are closed under reversal, each of the families, except the contextsensitive family, is closed under left quotient by a regular language, according to the second property above.
References
 1 J.E. Hopcroft, J.D. Ullman, Formal Languages^{} and Their Relation^{} to Automata, AddisonWesley, (1969).
Title  quotient of languages 

Canonical name  QuotientOfLanguages 
Date of creation  20130322 18:56:06 
Last modified on  20130322 18:56:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 68Q70 
Classification  msc 68Q45 
Defines  quotient 
Defines  left quotient 
Defines  right quotient 