Question

To determine the equivalence classes under the given relation \(R\).

Short answer

There are \(6\) equivalence classes of colorings as shown below:

\(\begin{array}{l}{(bbbb)_R} = \{ bbbb\} \\{(rrrr)_R} = \{ rrrr\} \\{(brrr)_R} = \{ brrr,rbrr,rrbr,rrrb\} \\{(rbbb)_R} = \{ rbbb,brbb,bbrb,bbbr\} \\{(bbrr)_R} = \{ bbrr,brrb,rrbb,rbrb\} \\{(brbr)_R} = \{ brbr,rbrb\} \end{array}\)

Step-by-step solution

Given data

The relation \(R\) on the set of all colorings of a \(2 \times 2\) chessboard, where a square may be arbitrarily colored red or blue.

Concept  used of Equivalence class

An equivalence class is defined as a subset of the form\(\{ x \in X:xRa\} \), where\(a\)is an element of\(X\)and the notation "\(xR{y^{\prime \prime }}\)is used to mean that there is an equivalence relation between\(x\)and\(y\).

Find the equivalence class

Each square can take on two colors: blue, b and red, r.

The equivalence classes from a checkerboard contains all checkerboards that can be obtained by rotating/reflecting the checkerboard.

Let \(XYZD\), where \(X\) represents the color of top left square, \(Y\) represents the color of top right square, \(Z\) represents the color of bottom left square and D represents the color of bottom right square.

The \(6\) (unique) equivalence classes of \(R\) then:

\(\begin{array}{l}{(bbbb)_R} = \{ bbbb\} \\{(rrrr)_R} = \{ rrrr\} \\{(brrr)_R} = \{ brrr,rbrr,rrbr,rrrb\} \\{(rbbb)_R} = \{ rbbb,brbb,bbrb,bbbr\} \\{(bbrr)_R} = \{ bbrr,brrb,rrbb,rbrb\} \\{(brbr)_R} = \{ brbr,rbrb\} \end{array}\)