Closure Properties

Question

What is difference between Σ*  and L*  ? 

 

Which is true ?

S1 :  Σ* – {ϵ} = Σ+

S2 :  L* – {ϵ} = L+ .

Comments

@Hemanth_13 i read it wrong

Take L = ${\varepsilon}$

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Thanks @Deepanshu @Shobhit Joshi

Is that the only case ??

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The following video solution covers ALL Variations, with Proofs:

https://youtu.be/nwIl4PxE8C8 

Answer 1

S1 :  Σ* – {ϵ} = Σ+  : TRUE  // Always true, definition of Σ+

S2 :  L* – {ϵ} = L+ . : FALSE // May or may not be true 

False when ϵ belongs L, then L+ and L* both will contain ϵ.

PS: In S2, it depends on given language purely.

 

Comments

Sir can you tell Me what does L* represent?

 I know that sigma * represent the set of all the strings possible over Sigma of length [0 ,......)

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@pradeepchaudhary Please check screenshots below,

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According to this screenshot L* will always contain epsilon so, the second statement has to be true...?

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L* will always contain epsilon, but L+ may or may not be contain.

See, if Given Language L is like {ϵ, aa, bb}  over {a,b} then L+ will also contain ϵ.

hope you can understand

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Second statement will be true then ??
L*-epsilon =L+

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No,