our friend claims that any language can be recognized by a finite automaton. Here

is their argument:

$($

i

$)$

$$A language is a set of strings. Any language A can be rewritten as A

$=$

$$S

w in A

$\{$

w

$\}$

$.$

$($

ii

$)$

$$For any string w

$,$

$\{$

w

$\}$

$$is a regular language since we can use the regular

expression w to specify a language that only contains w

$.$

$($

iii

$)$

$$By

$($

i

$)$

$$and

$($

ii

$)$

$,$

$$A can be represented as a union of regular languages.

$($

iv

$)$

$$The union of any two regular languages is also a regular language; that is

$,$

regular languages are closed under union.

$($

v

$)$

$$By

$($

iii

$)$

$$and

$($

iv

$)$

$,$

$$A itself must be regular as it is a union of regular languages.

$($

vi

$)$

$$Since A is regular, there exists some finite automaton that recognizes it

$.$

Therefore, any language has a finite automaton that recognizes it

$.$

Which steps of this argument are valid and which are not? Explain your answers.