no diagram is not neccessary for right quotient (L1/L2) by making differenr strings we can find it
The right quotient of L1 with L2 is the set of all strings x where you can pick some y from L2 and append it to x to get something from L1. That is, x is in the quotient if there is y in L2 for which xy is in L1.)
Let's agree to write the quotient of L1 by L2 as L1/L2.
Here are some examples:
- Say that L1 is the language {fish,dog,carrot} and that L2 is the language {rot}. Then L1/L2, the quotient of L1 by L2, is the language {car}, because car is the only string for which you can append something from L2 to get something from L1.
- Say that L1 is the language {carrot,parrot,rot} and that L2 is the language {rot}. Then L1/L2 is the language {car,par,ϵ}. Say that L3= {rot,cheese} Then L1/L3 is also {car,par,ϵ}
- Say that L1={carrot} and L2={t,ot}. Then L1/L2 is {carro,carr}.
- Say that L1={xab,yab}and L2={b,ab} Then L1/L2 is {xa,ya,x,y}
-
Say that L1={ϵ,a,ab,aba,abab,…} and L2={b,bb,bbb, }L2=Then L1/L2 is {a,aba,ababa,…}
-
In general, if L2 contains ϵ, then L1/L2 will contain L1
- In general, if L2=P∪Q, then
L1/L2=(L1/P)∪(L1/Q).
- In general, if L1=P∪Q then
L1/L2=(P/L2)∪(Q/L2).
- The two foregoing facts mean that you can calculate the right quotient of two languages L 1and L2 as follows: Let s1 be some element of L1 and let s2 be some element of L2. If s2 is a suffix of s1, so that s1=xs2 for some string x, then x is in the quotient L1/L2. Repeat this for every possible choice of s1and s2 and you will have found every element of L1/L2.