Q: What is meant by $ab$ on words $a$ and $b$ in $\{ab\ |\ a,b \in Σ^*\}$? Given language $L$ := $\{ab\ |\  a,b \in Σ^*\}$, $Σ := \{blue, green\}$.


*

*Is the notation "$ab$" above taken to be  word concatenation, such that $\{bluegreen\} \subset L$?

*What occurs when $L$ := $\{aa^R\ |\  a \in Σ^*\}$, given the same $Σ$, where $a^R$ is word reversal? What would the elements of $L$ be? Could you list them to elucidate what is meant by $aa^R$? If necessary, change $Σ$ to fit your answer.


A: A letter is the most basic element of a word.  It cannot be decomposed into more basic elements.  A letter is usually represented by one symbol such as $a$ or $b$.
An alphabet is a non-empty set of letters.  For example $\Sigma=\{a,b\}$ is the alphabet consisting of the two letters $a$ and $b$.
A word is a concatenation of letters from an alphabet, usually denoted by juxtaposing the letters, where the order of concatenation matters.  For example, $ab$ is a word from the alphabet given above, and it is not the same as $ba$.
Sometimes the concept of an empty word is useful.  The symbol $\epsilon$ is sometimes used to denote an empty word.
A word can be represented by more than one symbol.  For example, $ab$.
A word can also be represented by one symbol.  If the word is composed of one letter, then the symbol for the word is the same as the letter.  For example, the word $a$ is composed of the letter $a$.
If the word is composed of more than one letter, then a one-symbol representation for the word is usually a symbol that is not a letter of the given alphabet.  For example, $w=ab$ is a meaningful way of representing the word $ab$ if $w$ is not a letter of the given alphabet.
Words can also be concatenated.  For example, if $w=ab$ and $x=abb$, then $wx=ababb$, $wb=x$, and $\epsilon w=w\epsilon$.
The reversal of a word $w$, denoted by $w^R$, is the word formed by concatenating its letters in the reverse order.  For example, if $w=ab$ then $w^R=ba$.  The reversal of a letter is the letter itself.  For example $a^R=a$.
A word $w$ is a palindrome if $w=w^R$.  Note that $ww^R$ is a palindrome, but not all palindromes are of this form.  For example, $aba$ is a palindrome but is not of the form $ww^R$.
You defined $\Sigma$ to be the set of the two letters $blue$ and $green$.  This is very unusual.  If you insist on using this notation, then one possible word for this alphabet is $bluegreen$; its reversal is $greenblue$.
