Let S be a set closed under a binary operation *.
Let A and B be disjoint subsets of S such that [tex]A \cup B = S[/tex] and for any [tex]a_{1}, a_{2}, a_{3} \in A[/tex] and [tex]b_{1}, b_{2}, b_{3} \in B[/tex], we have that [tex]a_{1} * a_{2} * a_{3} \in A[/tex] and [tex]b_{1} * b_{2} * b_{3} \in B[/tex].
Prove that A and B are also closed under *.
Should I post a hint?
I think you should post a problem that doesn't use set notation :P