Let Σ={a,b,c}. For x ∈ Σ∗, and α ∈ Σ, let #α(x) denote the number of occurrences of αα in x. Which one or more of the following option(s) define(s) regular language(s)?

The correct option is:
{a^mbⁿ | m, n ≥ 0}

Step-by-step Explanation:

1. Definition of a Regular Language
A language is defined as regular if it can be recognized by a Finite Automaton (such as a DFA or NFA) or, equivalently, if it can be expressed using a regular expression. Finite automata have a finite amount of memory, which means they can count modulo a fixed integer but cannot compare or match the counts of two different independent sets of symbols if those counts can grow arbitrarily large.

2. Analysis of the Correct Option
Let us consider the given language:
$L=\{a^m{b}^n\mid m,n\ge 0\}$
In this language, a sequence of $a$'s is followed by a sequence of $b$'s.
Key observation: The power of $a$ (which is $m$) and the power of $b$ (which is $n$) are completely independent of each other. There is no constraint requiring a comparison between $m$ and $n$ (for instance, we do not require $m=n$ or $m<n$).
Consequently, a machine does not need to remember the number of $a$'s it has read when it starts reading $b$'s.

We can write a regular expression for this language directly:
$a^{*}{b}^{*}$
Because the language can be defined by a regular expression and recognized by a simple two-state DFA, it is a regular language.