What does this language notation specify? I am given this exercise:
Let L1 ={akbk : k &gt; 0} and L2={ck : k &gt; 0}. For each of the following strings wi, state and explain whether or not wi ∈ L1L2.
w1=ε
w2=aabbcc
w3=abbccw
w4=aabbcccc
w5=ccaabb
w6=ab
So far I haven't seen such a notation for languages as the one that L1 and L2 are seen in. For example, I had seen other notations such as: L1={w∈{a,b}∗: every a in w is immediately preceded and followed by a b} which I could understand. A parallel between the two notations will be much appreciated.
Moreover, what does it refer by L1L2? Is this counted as a union or a concatenation?
What I could figure out from this exercise is what w1 is surely not in L1L2, as k must be over 1. Following the same logic, the other answer would be w2 as k will always be the same in both L1 and L2, hence the number of a's, b's and c's shouldn't differ.
Is the above thought correct or is it more to this notation than I have already thought of?
Language concatenation is defined as follows:
$$
L_1 L_2 = \{ w_1 w_2 : w_1 \in L_1, w_2 \in L_2 \}.
$$
In other words, a word $w$ belongs to $L_1 L_2$ if it can be written as the concatenation of two words, $w_1 \in L_1$ and $w_2 \in L_2$.
This definition should suffice for solving the exercise.