空字符串是所有字母的子集还是元素？

Question

I'm having trouble understanding when the empty string (epsilon) is a subset or element of an alphabet? My understanding was that epsilon was only part of a language, but my TA in the class said it was an element of all alphabets so now I am confused.

e.g. would {a,b,c} contain epsilon as an element?
e.g. would {} contain epsilon as an element?
e.g. is {eps} a subset of all alphabets or languages?

Answer 1

This question is probably better suited for cs.stackexchange but I will try to help you according to my understanding, please do correct me if necessary.

In general, your own intuition seems quite correct to me. ϵ is not automatically part of every alphabet. It is the empty string of characters.

However, this means that ϵ is a string over any alphabet , even your alphabet {a, b, c} .

So to answer your three examples:

1. No, it does not. If {a, b, c} is an alphabet, it is a set of symbols, and ϵ is a string. However, ϵ is definitely part of some languages defined over this alphabet.

2. No, {} is the empty set, and it contains nothing, not even ϵ .

3. {ϵ} is the set containing only ϵ . ϵ is a string, not a symbol, so it is not a subset of all alphabets (however, it seems there are cases where some alphabets are defined to contain ϵ but that is a different confusing story). It is also not a subset of all languages, because consider the language L = {aa, ab, ba, bb} . The empty string ∈ is not one of these elements.

The analogue to set theory might cause some additional confusion. Notice that the empty set is a subset of every set. The empty string is not a subset of every language, but rather it is a substring of any string.