The question is as follows:

[Concerning the square embedded in the plane,] prove that the $90^{\circ}$ clockwise rotation $\sigma$ and the reflection through the north/south axis $\rho$ do not commute.

I'm generally just confused on how one would go about proving something like this. I have represented the square by numbering it's vertices as $\{1,2,3,4\}$ and I can show that the composition of transformations do not equal each other, but I have a feeling this is insufficient. I would really appreciate some guidance on the form a proof like this might take. Thanks a bunch!


Your approach is fine. If, as you wrote, the vertices are $1$, $2$, $3$, and $4$ (labelled clockwise), and your reflection $r$ is with respect to the line defined by $1$ and $3$, and your rotation is called $\rho$, then

  • $\rho\bigl(r(1)\bigr)=\rho(1)=1$;
  • $r\bigl(\rho(1)\bigr)=r(2)=3$.

Since $\rho\bigl(r(1)\bigr)\neq r\bigl(\rho(1)\bigr)$, then $\rho\circ r\neq r\circ\rho$. So, $\rho$ and $r$ do not commute.

  • $\begingroup$ obg, but is there not a more rigorous way to do this other than just through examples? $\endgroup$ – Gwen Di Feb 23 at 20:57
  • $\begingroup$ What is there about this approach which is not rigorous? Proving that $\rho\circ r\neq r\circ\rho$ means proving that $\rho\bigl(r(k)\bigr)\neq r\bigl(\rho(k)\bigr)$, for some $k\in\{1,2,3,4\}$. $\endgroup$ – José Carlos Santos Feb 23 at 21:00
  • 3
    $\begingroup$ @gwen The way to disprove a statement is to give a special example called a counterexample. Perhaps you are confusing this with trying to prove statements with examples? $\endgroup$ – rschwieb Feb 23 at 23:05
  • $\begingroup$ @JoséCarlosSantos: did you mean $\rho \circ r$ where you wrote $\rho \circ c$? $\endgroup$ – J. W. Tanner Feb 24 at 2:51
  • $\begingroup$ @J.W.Tanner Yes! I've edited my answer. Thank you. $\endgroup$ – José Carlos Santos Feb 24 at 8:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.