so i have this description: S consists of all strings of a's, b's and c's where an increasing number of a's come before an increasing number of b's and c's or c's and b's. does this make sense for this recursive definition of the set S over {a,b,c}: 1. a is in S and 2. if x in S then ax, xb, xc are in S?
Yes, the given recursive definition for the set S over {a, b, c} makes sense for the given description.
Let's break down the recursive definition to understand how it aligns with the description of the set S:
1. "a is in S": This statement means that the string "a" is an element of the set S. As per the description, strings in S consist of a's, b's, and c's, and it is mentioned that an increasing number of a's come before an increasing number of b's and c's. So, the presence of a single "a" in the set S is consistent with the description.
2. "if x in S, then ax, xb, xc are in S": This statement describes the recursive aspect of the definition. It states that if a string x is in S, then the strings obtained by appending an "a" (ax), a "b" (xb), and a "c" (xc) to x are also in S.
This recursive definition captures the pattern of an increasing number of a's coming before an increasing number of b's and c's or vice versa, as stated in the description. By recursively applying the second rule (2), we can generate more strings in S that adhere to the pattern.
For example, using the second rule (2) on the string "a" (which we know is in S due to the first rule (1)), we can generate the following strings: "aa", "ab", and "ac". Applying the rule again on these new strings will generate even more strings that follow the desired pattern.
Hence, the recursive definition aligns with the description of the set S, and it is indeed a valid definition for the given description.