If L is a language, how is L* related to L*L*? (discrete mathematics) If L is a language, how is L* related to L* L*?
I understand that the * represents all possible combinations of the elements of L (infinite).
However, L* L* would be tuples of combinations of the elements of L*, right?  If so, and if we assume null or an empty character exists, then does that make L* a subset of L* L*?  Or is it rather, an infinite set of elements is equal to an infinite set of elements?
 A: If $L_1$ and $L_2$ are languages, $L_1L_2=\{xy:x\in L_1\text{ and }y\in L_2\}$: it’s the set of compositions of words in $L_1$ with words in $L_2$. Unless you think of the words themselves as tuples, there are no tuples involved. 
In the case of $L^*L^*$, note that $\lambda\in L^*$ (where $\lambda$ is the empty word; you may be used to calling it $\epsilon$), and $x\lambda=x$ for each $x\in L^*$, so it is indeed true that $L^*\subseteq L^*L^*$: every word in $L^*$ is also in $L^*L^*$. But we can say more: in fact $L^*=L^*L^*$. 
To see this, suppose that $xy\in L^*L^*$, where $x,y\in L^*$. If $x=\lambda$, then $xy=y\in L^*$, and similarly, if $y=\lambda$, then $xy=x\in L^*$, so suppose that $x\ne\lambda\ne y$. Then there are positive integers $m$ and $n$ and words $u_1,\ldots,u_m,v_1,\ldots,v_n\in L$ such that $x=u_1u_2\ldots,u_m$ and $y=v_1v_2\ldots v_n$. But then
$$xy=u_1u_2\ldots u_mv_1v_2\ldots v_n\in L^*\;,$$
so in every case we find that $xy\in L^*$. Thus, $L^*L^*\subseteq L^*\subseteq L^*L^*$, and it follows that $L^*L^*=L^*$.
A: $L^* = \bigcup_{n \geq 0} L^n$, with $L^0 := \{\epsilon\}$, where $\epsilon$ the empty word. Therefore $L^* = L^*L^*$, as $L^* = \epsilon L^*$, we have that $L^* \subset L^*L^*$, and clearly $L^*L^* \subset L^*$.
