MHB Equivalence Classes of Bracelet Beads: Red, White, Blue

AI Thread Summary
The discussion focuses on defining equivalence classes for bracelets made of three beads, each colored red, white, or blue. The relation R identifies two bracelets as equivalent if one can be obtained from the other through rotation or reflection. Participants clarify that each equivalence class corresponds to a unique bracelet configuration. Examples of equivalence classes include combinations like (red, red, red) and (red, red, blue) along with its rotations and reflections. The goal is to determine all unique bracelet configurations based on these transformations, emphasizing the importance of accounting for symmetry in the classification process.
linag96
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Each bead on a bracelet with three beads is either red,
white, or blue.
Define the relation R between bracelets as: (B1, B2),
where B1 and B2 are bracelets, belongs to R if and only
if B2 can be obtained from B1 by rotating it or rotating it
and then reflecting it.
What are the equivalence classes of R?

I'm a little lost on how to make these classes, is it just something like
(red, blue, white), (red, white, blue), (blue, red, white), (blue, white, red) etc?
Thank you for your help.
 
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linag96 said:
Each bead on a bracelet with three beads is either red,
white, or blue.
Define the relation R between bracelets as: (B1, B2),
where B1 and B2 are bracelets, belongs to R if and only
if B2 can be obtained from B1 by rotating it or rotating it
and then reflecting it.
What are the equivalence classes of R?

I'm a little lost on how to make these classes, is it just something like
(red, blue, white), (red, white, blue), (blue, red, white), (blue, white, red) etc?
Thank you for your help.

Hi linag96! ;)

Yes it is.
For instance {(red,red,red)} is an equivalence class.
And so is {(red,red,blue), (red,blue,red), (blue,red,red)}.
Which would the other classes be? (Wondering)

Btw, note that each equivalence class corresponds to a unique bracelet.
We are effectively finding the number of unique bracelets.

\begin{tikzpicture}
%preamble \usetikzlibrary{arrows,shadows}
[>=stealth',font=\large,scale=1,very thick,
button/.style 2 args={%
circle,
minimum size=0.6cm,
top color=#1!40!white,
bottom color=#1!60!black,
draw=#1!90!black,
thick,
general shadow={%
shadow xshift=.4ex, shadow yshift=-.4ex,
opacity=.5, fill=black!50,
}
}]
\newcommand{\bracelet}[4]
{%
\draw #1 node[button={#2}{}] (A) {1}
+(1,-0.5) node[button={#3}{}] (B) {2}
+(2,0) node[button={#4}{}] (C) {3};
\draw[bend right] (A) edge (B) (B) edge (C) (C) edge (A);
}

\node at (-1,0) {\LARGE $\{$};
\bracelet{}{red}{red}{red};
\node at (3,0) {\LARGE $\}$};

\node at (-1,-2) {\LARGE $\{$};
\bracelet{(0,-2)}{red}{red}{green};
\bracelet{(3,-2)}{red}{green}{red};
\bracelet{(6,-2)}{green}{red}{red};
\node at (9,-2) {\LARGE $\}$};

\end{tikzpicture}
 
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