Give an example of languages L1 and L2 that satisfy the conditions (a)
L1L2=L2L1, (b) L2 is a proper nonempty subset of L1, and (c) L2≠{ε}.
Your subject is NOT the name of your school.
Ya sorry the subject code is CSCI 119. Finite Automate but this sum is of Regular Languages
Anyone?
To find an example of languages L1 and L2 that satisfy the given conditions, we need to consider a few things:
Condition (a) states that L1 concatenated with L2 must be equal to L2 concatenated with L1.
Condition (b) states that L2 should be a proper nonempty subset of L1, meaning L2 should be a subset of L1 but not equal to L1.
Condition (c) states that L2 should not be equal to the empty string ε.
To satisfy condition (a), we need to find languages where the order of concatenation does not matter. One such example is the set of all binary strings: L1 = {0, 1}* and L2 = {0, 1}+. Here, the "*" represents the Kleene star and means any number of repetitions (including zero) of the elements within the braces, and "+" represents at least one repetition.
Now let's check if the conditions are satisfied:
Condition (a): L1L2 = L2L1
In this case, L1L2 represents all possible binary strings followed by at least one binary digit: {0, 1}*{0, 1}+ = {0, 1}+{0, 1}*.
L2L1 represents at least one binary digit followed by all possible binary strings: {0, 1}+{0, 1}* = {0, 1}*{0, 1}+.
Therefore, condition (a) is satisfied.
Condition (b): L2 is a proper nonempty subset of L1
L1 = {0, 1}* includes all possible binary strings (including the empty string ε), but L2 = {0, 1}+ does not include the empty string ε.
Therefore, L2 is a proper nonempty subset of L1, satisfying condition (b).
Condition (c): L2≠{ε}
In this case, L2 = {0, 1}+ does not include the empty string ε, so condition (c) is met.
Hence, the example of languages L1 = {0, 1}* and L2 = {0, 1}+ satisfies all the given conditions.