What language does the following REGEX r represent?

r = ( lam* )*

Note: phi is used for
𝜙
, and lam is used for
𝜆

Question

What language does the following REGEX r represent?

r = ( lam* )*

Note: phi is used for 
𝜙
, and lam is used for 
𝜆

BlackTom AI · Accepted Answer

Since lam denotes the empty string λ, treat lam as the string λ. The Kleene star L* of a language L is the set of all finite concatenations of strings from L (including the length-0 concatenation, which is λ).

Step 1 — compute lam*:
- lam* = { (lam)^k | k ≥ 0 }.
- For every k ≥ 0, (lam)^k = λ (because concatenating the empty string any number of times yields the empty string).
- Therefore lam* = {λ} (a singleton language whose only element is the empty string).

Step 2 — compute (lam*)*:
- Now treat lam* = {λ}. Apply star again: (lam*)* = {λ}* = { (λ)^k | k ≥ 0 }.
- As before, (λ)^k = λ for all k, so {λ}* = {λ}.
- Thus (lam*)* = {λ}.

Why the correct answer is { lam }:
- The language denoted by r = ( lam* )* is exactly the singleton set containing the empty string, i.e., {λ}. In the problem’s notation, that is written { lam }.

Why the other options are incorrect:
- "{ }" usually denotes the empty set ∅ (no strings). Our language contains λ, so it is not empty.
- "{ phi }" would denote a set containing φ (the empty language) or be interpreted as something involving the empty set symbol; in any standard interpretation this is not equal to the singleton {λ}. φ (empty language) is not a string, so {φ} is not {λ}.
- "None is correct." is false because one of the provided options does match the result.

Conclusion: (lam*)* = {λ}, so the correct choice is "{ lam }", which matches the provided answer.

What language does the following REGEX r represent? r = ( lam* )* Note: phi is used for 𝜙 , and lam is used for 𝜆Single choice

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