I was studying binary operation on a set. Then the following question came to mind. I tried to find an answer. also searched in website but could not get any satisfactory answer.
the question is: is it possible to define two distinct binary operation on a same set?
the reason to search an answer for it is, if yes, then I think we can construct different group structure on a same set.
But thats the main problem. How to create a new binary operations from an existing one? I am searching for some recreational answer. please help me. In case it is already solved, kindly provide me the link
thanks in advance
There are many groups which have the same cardinality. Suppose you have two groups $(G,\circ)$ and $(H,\cdot)$ and an arbitrary (set-theoretic) bijection $\varphi : G \to H$. You can define a new multiplication $\star$ in $G$ as follows : $$ g_1 \star g_2 \overset{def}= \varphi^{-1}( \varphi(g_1) \cdot \varphi(g_2)). $$ and $\star$ doesn't have to coincide with the multiplication in $G$, i.e. $\circ$. For instance, if $(G,\circ)$ and $(H,\cdot)$ are not isomorphic groups, then since $(G,\star)$ is isomorphic to $(H,\cdot)$ (the isomorphism being given by $\varphi : (G,\star) \to (H,\cdot)$), you see that $(G,\circ)$ and $(G,\star)$ are not isomorphic groups, so of course the binary operations $\star$ and $\circ$ have to be different.
Hope that helps,