In mathematics and computer science, the right quotient (or simply quotient) of a language L_1 with respect to language L_2 is the language consisting of strings w such that wx is in L_1 for some string x in L_2. Formally: In other words, for all the strings in L_1 that have a suffix in L_2, the suffix is removed. Similarly, the left quotient of L_1 with respect to L_2 is the language consisting of strings w such that xw is in L_1 for some string x in L_2. Formally: In other words, we take all the strings in L_1 that have a prefix in L_2, and remove this prefix. Note that the operands of \backslash are in reverse order: the first operand is L_2 and L_1 is second.

Example

Consider and Now, if we insert a divider into an element of L_1, the part on the right is in L_2 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 a^n b^{n-i} or ; and L_1 / L_2 can be written as

Properties

Some common closure properties of the quotient operation include: These closure properties hold for both left and right quotients.