In mathematics and computer science, the right quotient (or simply quotient) of a language
1
with respect to language
is the language consisting of strings w such that wx is in
for some string x in
[1] Formally:
In other words, for all the strings in
that have a suffix in
, the suffix is removed.
Similarly, the left quotient of
with respect to
is the language consisting of strings w such that xw is in
for some string x in
Formally:
In other words, we take all the strings in
that have a prefix in
, and remove this prefix.
Note that the operands of
are in reverse order: the first operand is
Now, if we insert a divider into an element of
, the part on the right is in
only if the divider is placed adjacent to a b (in which case i ≤ n and j = n) or adjacent to a c (in which case i = 0 and j ≤ n).
The part on the left, therefore, will be either
{\displaystyle a^{n}b^{n-i}}
{\displaystyle a^{n}b^{n}c^{n-j}}
can be written as
Some common closure properties of the quotient operation include: These closure properties hold for both left and right quotients.