Which of the following statements are true regarding Pumping lemma for regular languages?

• I. If a language is regular, then it always satisfies pumping lemma.
• II. If a language is satisfies pumping lemma, then it is always regular language.
• III. Pumping lemma is used to show a language is regular.
• IV. Pumping lemma is used to show a language is non-regular.

(A) I and III
(B) II and IV
(C) II and III
(D) I and IV


Answer: (D)

Explanation: Pumping lemma is for negativity test. If a language contradicts pumping lemma then it is not a regular language. If a language is satisfies pumping lemma, then it may or may not be regular language.

1. Pumping Lemma: ¬q → ¬p where, q is pumping lemma and p is regular language. It is Contrapositive that means if a language does not satisfies pumping lemma, then it can not be regular language. It is always true.
2. Then, it is also correct that p → q. It is implication that means if a language is regular then it is always satisfies pumping lemma.

Also, note that its inverse ¬p → ¬q and converse q → p need not be true according to prepositional logic.

So, option (D) is correct.

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