If we have a language $L = \{0\}^*$ over the alphabet $\Sigma=\{0,1\}$, what is $\Sigma^*\backslash L$?
That's what I think:
$\{0,1\}^* = \{\epsilon, 0, 1, 00, 01, 10, 11, 000, 001, ... \}$
$\{0\}^* = \{\epsilon, 0, 00, 000, 0000, ... \}$
$\{0,1\}^*\backslash \{0\}^*= \{1,01,10,11,001,...\}$
In other words, the resulting set should contain everything from $\Sigma^*$ except the empty string and any string that contains 0s but not 1s.
If my thinking is correct, then how can I express $\{1,01,10,11,001,...\}$ more succinctly using set notation? I was thinking of something like $\{0,1\}^*\{1\}\{0,1\}^*$ but I'm not sure if it's correct or if there's a better way.