If L is a regular language and F is a finite language (i.e., a language with a finite number of words), then (L U F) must be a regular language Group of answer choices True False
The correct answer and explanation is:
Correct Answer: True
Explanation:
To determine whether the union of a regular language LL and a finite language FF is regular, we rely on foundational concepts from formal language theory, particularly the closure properties of regular languages.
Regular Languages and Closure Properties:
A regular language is one that can be expressed using a regular expression, or recognized by a finite automaton (DFA or NFA). One of the key characteristics of regular languages is that they are closed under union. This means that if L1L_1 and L2L_2 are both regular, then L1∪L2L_1 \cup L_2 is also regular.
But what if only one of them is regular and the other is just finite?
Finite Languages Are Regular:
A finite language is a language that contains a finite number of strings. Any finite language is automatically regular because we can construct a finite automaton that accepts exactly the strings in that language. For instance, if F={w1,w2,…,wn}F = \{w_1, w_2, …, w_n\}, then we can design a finite automaton with a separate path for each word wiw_i, accepting only those strings. Alternatively, we can express FF with the regular expression w1∪w2∪…∪wnw_1 \cup w_2 \cup … \cup w_n, which is clearly a regular expression.
Thus, every finite language is regular.
Applying Closure Under Union:
Given:
- LL is regular.
- FF is finite ⇒F\Rightarrow F is also regular.
Since both LL and FF are regular, and regular languages are closed under union, L∪FL \cup F is regular.
Conclusion:
The union of a regular language and a finite language is regular. Therefore, the statement:
“If LL is a regular language and FF is a finite language, then L∪FL \cup F must be a regular language.”
is True.