If L is a language defined by :
L = { awa | w ∈ {a, b}* },
is aa a string of the language L? (notice that w is being null string here)
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According to you definition it seems L consists only of two words
abaandaaa.EDIT: after you have edited the question I can say
yes "aa" is a word of this languagew ∈ {a, b}*means zero or more chars of this alphabet and thus w may have zero chars and be empty.