The following question is taken from Arrows, Structures and Functors by Arbib and Manes.
For each set $A$, the set $A*$ of all 'words on the alphabet $A$' may be defined as
$$A*=\coprod_{n=0}^{\infty}A^n$$
where $A^0=\{\land\}$ 'the empty word', $A^{n+1}=A^n\times A$). Why can't we use an ordinary union" [Hint: Construct a set $A$ such that $A\cap(A\times A) \neq \emptyset$]
I try to create the non empty set in the hint, but this is my result:
$\text{If we have the alphabet}$ $\sigma=\{0,1\}$
and set $A=\{00, 01\}$, then for $n=2,$
$A\times A=A^2=\{0000, 0001, 0100, 0101\},$ $\text{don't we then have}$
$A\cap(A\times A)=\emptyset$, but with $A \cup (A\times A)=\{00, 01, 0000, 0001, 0100, 0101\}$
I am obviously doing something wrong. My knowledge about formal language and automata are almost non existent other than knowing how to do rudimentary set theoretic operations with them. Can someone give me some help me solving this question and also point out where I am going wrong or points my misunderstanding or insufficient understanding.
Thank you in advance.