Consider the languages \(L_{1}= \phi\) and \(𝐿_2={1}\). Which one of the…
2017
Consider the languages \(L_{1}= \phi\) and \(𝐿_2={1}\). Which one of the following represents \(L_{1}^{\ast}\cup L_{2}^{\ast} L_{1}^{\ast}\)?
- A.
\(\{∈\}\) - B.
\(\{∈,1\}\) - C.
\(𝜙\) - D.
\(1^∗\)
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Correct answer: D
Key facts: compute each component and simplify.
L1 = ∅, so L1* = {ε} (only the empty string from zero concatenations).
L2 = {1}, so L2* is the set of all strings of zero or more 1's, i.e. {1}* (this set already contains ε).
Concatenate: L2* L1* = {1}* {ε} = {1}* (concatenating with {ε} leaves the set unchanged).
Union: L1* ∪ (L2* L1*) = {ε} ∪ {1}* = {1}* (since ε is already in {1}*).
Answer: {1}* — the set of all strings consisting of zero or more 1's (including the empty string).
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