The operation DM(L) is defined as follows:
1. Throw away every even-length string from L.
2. For each odd-length string, remove the middle character.
For example, if L = {001, 1100, 10101}, then DM(L) = {01, 1001}. That is, even-length string 1100 is deleted, the middle character of 001 is removed to make 01, and the middle character of 10101 is removed to make 1001.
It turns out that if L is a regular language, DM(L) may or may not be regular. For each of the following languages L, determine what DM(L) is, and tell whether or not it is regular.
• L1: the language of regular expression (01)*0.
• L2: the language of regular expression (0+1)*1(0+1)*.
• L3: the language of regular expression (101)*.
• L4: the language of regular expression 00*11*.
Now, identify the true statement below.
a) DM(L4) is regular; it is the language of regular expression 0*1*.
b) DM(L1) is not regular; it consists of all strings of the form (01)n(00)(10) n.
c) DM(L1) is not regular; it consists of all strings of the form (01)n(00+ε) (10)n.
d) DM(L3) is not regular; it consists of all strings of the form (101)n(00+11) (101)n.
Could you please explain.I do not understant how to solve?
