# Intersection between language and regular set

This is tangentially related to a question I asked on the math stackexchange.

I read in a proof that if

\begin{align*} L_1 &= \{w ∈ \{a, b, c\}^* : \text{w has the same number of a, b, and c} \}, \\ R &= L(a^* b^* c^*), \end{align*}

then $L ∩ R = \{a^nb^nc^n: n ≥ 0\}$.

Now unless I'm fundamentally misunderstanding intersection, why is the intersection between this language and the regular set is $a^nb^nc^n$ and not something like $a^ib^jc^k$ where $i,j,k \geq 0$? Is $R$ not the set including strings like abbccc or aaabbc?

• A word like $abbccc$ would only be in the intersection of $L1$ and $R$ if it was in both $L1$ and in $R$. $abbccc$ is not in L1, because it has a different number of a's, b's and c's (1 a, 2 b's and 3 c's). May 7 '17 at 7:45
• @BoltonBailey and here I was looking up the properties of the kleene star. thanks May 7 '17 at 7:48
• @BoltonBailey Make an answer? May 7 '17 at 11:49

Because $A\cap B = \{x\mid x\in A\text{ and }x\in B\}$.