I'm having some problem with this question involving formal languages: let $L = \{ \epsilon, 01, abc \}$ and $T = \{x, y, z\}$. I'm supposed to find $L^3$, $LT$ and $TL$.
Now I know that the first one is just the set of all words length $3$ over the alphabet $L$, however, $I$ am confused by the presence of the empty word.
Would $L^3 = \{\epsilon \epsilon \epsilon, \epsilon \epsilon 01, \epsilon \epsilon abc, \epsilon 01abc, 01 \epsilon \epsilon, 0101 \epsilon, 010101, 01abc \epsilon, 01 \epsilon abc, \dots\}$
or
would $L^3 = \{010101, 0101abc, 01abc01, abc0101, \dots\}$?
Your first answer is correct, except that you should simplify the words containing $\epsilon$: $\epsilon\epsilon\epsilon=\epsilon$, $01\epsilon abc=01abc$, and so on. In fact, you can eliminate every instance of $\epsilon$ except the empty word itself, because for any word $w$ you know that $w\epsilon=\epsilon w=w$.