language

a set of strings all of which are chosen from some $\text{[math]}$ , where $\text{[math]}$ is a particular alphabet, is called a language. if $\text{[math]}$ is an alphabet, and $\text{[math]}$ , then $\text{[math]}$ is a language over $\text{[math]}$ . notice that a language over $\text{[math]}$ need not include strings with all the symbols of $\text{[math]}$ , so once we have established that $\text{[math]}$ is a language over $\text{[math]}$ , we also know it is a language over any alphabet that is a superset of $\text{[math]}$ .
[cite:@john_automata_2006]

since languages are sets, set operations are applicable to them.

$\text{[math]}$ , the reverse of $\text{[math]}$ is defined by $\text{[math]}$ , where $\text{[math]}$ denotes the reverse of the word $\text{[math]}$ .

$\text{[math]}$

prove/disprove

$\text{[math]}$
$\text{[math]}$

counter example for 1: $\text{[math]}$
proof for 2: $\text{[math]}$

given $\text{[math]}$ , we define $\text{[math]}$ recursively by $\text{[math]}$ .