# Can an alphabet for a Turing machine contain subsets of other alphabets?

For example;

Is {0,1,{a,b,c},d,e} a valid alphabet to form a language over and is it usable in any context?

An alphabet is a set of symbols, therefore if your you treat $$\{a,b,c\}$$ as a single symbol (in some other alphabet), it is a valid alphabet.
• @WeCanBeFriends Exactly like any other alphabet. $\{a,b,c\}$ is just another symbol, so it can be written on a single tape cell, but none of the $a, b$ or $c$ can be written individually, the same way you cannot write just a half of $0$ or any part of any other symbol. – Gogis Aug 3 '19 at 16:17
• What is the advantage to having {a.b,c} in the alphabet, as opposed to another symbol k? I'm not sure of a situation where it would be useful to have this set – WeCanBeFriends Aug 3 '19 at 16:20
• @WeCanBeFriends Random example from the top of my head: Imagine you would need to construct a TM that would be given a set of 3 numbers from the range $0-100$ and accept iff their average would be $> 47$. It would be more convenient to describe the TM using an alphabet where each input would be represented by a single symbol, because that way the TM would only need 2 states. – Gogis Aug 3 '19 at 17:28