One of the defining properties of regular languages is that they're recognized by DFAs.

a) Write "every regular language can be recognized by some DFA" as a first-order formula, and find its complement, contrapositive, and dual. Express each of these as both a first-order formula and a written description in English. Hint: for this example, you can treat "can be recognized by some DFA" as a simple predicate

. b) What can we infer about nonregular languages from this?

c) What if we didn't treat "can be recognized by some DFA" as a predicate, but instead wrote it as its own expression in first-order logic?