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For Σ = {𝑎, 𝑏}, let us consider the regular language 𝐿 = { 𝑥 |𝑥 = 𝑎2+3𝑘 or 𝑥 = 𝑏10+12𝑘, 𝑘 ≥ 0}. Which one of the following

Q. For Σ = {𝑎, 𝑏}, let us consider the regular language 𝐿 = { 𝑥 |𝑥 = 𝑎2+3𝑘 or 𝑥 = 𝑏10+12𝑘, 𝑘 ≥ 0}. Which one of the following can be a pumping length (the constant guaranteed by the pumping lemma) for 𝐿 ? (A) 3 (B) 5 (C) 9 (D) 24 Ans: 24 Solution: According to Pumping lemma, ...

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Let 𝐺 be an arbitrary group. Consider the following relations on 𝐺

Q. Let 𝐺 be an arbitrary group. Consider the following relations on 𝐺: 𝑅1: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅1𝑏 if and only if ∃𝑔 ∈ 𝐺 such that 𝑎 = 𝑔−1𝑏𝑔 𝑅2: ∀𝑎, 𝑏 ∈ 𝐺, 𝑎 𝑅2𝑏 if and only if 𝑎 = 𝑏−1 Which of the above is/are equivalence relation/relations? (A) 𝑅1 and 𝑅2 (B) 𝑅1 only (C) 𝑅2 only (D) ...

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If 𝐿 is a regular language over Σ = {𝑎, 𝑏}, which one of the following languages is NOT regular ?

Q. If 𝐿 is a regular language over Σ = {𝑎, 𝑏}, which one of the following languages is NOT regular ? (A) 𝐿 ⋅ 𝐿𝑅 = {𝑥𝑦 | 𝑥 ∈ 𝐿, 𝑦𝑅 ∈ 𝐿} (B) {𝑤𝑤𝑅 | 𝑤 ∈ 𝐿} (C) Prefix (𝐿) = {𝑥 ∈ 𝛴∗|∃𝑦 ∈ 𝛴∗ such that 𝑥𝑦 ∈ 𝐿} (D) Suffix (𝐿) = {𝑦 ∈ 𝛴∗|∃𝑥 ∈ 𝛴∗ such that 𝑥𝑦 ∈ 𝐿} Ans:{𝑤𝑤𝑅 | 𝑤 ∈ 𝐿} ...

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Let 𝑈 = {1,2, … , 𝑛}. Let 𝐴 = {(𝑥, 𝑋)|𝑥 ∈ 𝑋, 𝑋 ⊆ 𝑈}. Consider the following two statements on |𝐴|

Q.Let 𝑈 = {1,2, … , 𝑛}. Let 𝐴 = {(𝑥, 𝑋)|𝑥 ∈ 𝑋, 𝑋 ⊆ 𝑈}. Consider the following two statements on |𝐴|. I.           |𝐴| = 𝑛2𝑛−1 II.         |𝐴| = ∑𝑛𝑘=1     𝑘(𝑛)            Which of the above statements is/are TRUE? (A) Only I(B) Only II (C) Both I and II(D) Neither I nor IIAns: Both I and ...

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