Question

Show that for any language L, L* = (L*)* = (L+)* = (L*)+

Answer 1

Explanation:

∀L,L∗=(L∗)∗

I've started with

def.L∗=⋃i∈NLi,L0={ϵ},L1={L},Li+1={uv|u∈Li,v∈L}

then(L∗)∗=⋃i∈NL∗i,L∗0={ϵ},L∗1={L∗},L∗i+1={uv|u∈L∗i,v∈L∗}

L∗L∗0∈(L∗)∗=L∗ϵ∗=L∗ ∴L∗⊆(L∗)∗

L∗L∗0=L∗,L∗L∗1=L∗(def),L∗L∗i+1=L∗iL∗=L∗L∗ but the * operator is closed under concatenation, thus, L∗L∗∈L∗∴(L∗)∗⊆L∗

∴L∗=(L∗)∗

I simply don't have an intuition on whether this attempt at a proof is correct and I would appreciate any insight about it since the subsequent problems are to show L+=(L+)+ and then to argue whether (L∗)+=(L+)∗