Let L = {a, b}. For the language below, give an example of a string in the language, and a string not in the language.
{w is in L*: uvw = wvu, for some {u, v} in L*}
- To simplify, I would say that any string is in the language if u = w and v can be anything. Would that statement be correct?
-For example, a string in the language would be w = ab, where u = w = ab and v = bb. (But v can be anything as long as it is in L*, right?)(And by "give an example of a string", does the string have to contain more than one character or do "a" and "b" count as strings?)
To determine whether a string is in the given language, we need to satisfy the condition that for some u, v in L*, the concatenation uvw is equal to the concatenation wvu. Let's break down the language and answer your questions step by step:
1. The language L consists of the characters 'a' and 'b'.
2. For the language {w is in L*: uvw = wvu, for some {u, v} in L*}, the statement you made is partially correct. In fact, any string is in the language if u = w = v. However, it's important to note that u and v can also be empty, so w doesn't have to equal u. So, we can say any string where u = v and both are in L* is in the language.
3. To give an example of a string in the language, let's take w = "ab". In this case, we have u = w = ab and v = empty string (since u = v). Thus, the string "ab" is in the language because it satisfies the condition.
4. For a string that is not in the language, we need to find an example where uvw is not equal to wvu for any u and v in L*. Let's consider the string w = "aa". We would need to find u and v such that uvw = wvu. However, no matter what u and v we choose from L*, we cannot rearrange the characters to make both sides equal. Thus, the string "aa" is not in the language.
As for your question about the length of the string, any string, whether it contains a single character or multiple characters, can be considered an example. So, "a" and "b" can both be valid examples of strings in this language.
I hope this clarifies the language and helps you understand how to determine whether a string is in the given set.