Let A be the set of points in the rectangle with x and y coordinates between 0 and 1

Chapter 8, Problem 33

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Let A be the set of points in the rectangle with x and y coordinates between 0 and 1. That is,

\(A=\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid 0 \leq x \leq 1 \quad \text { and } \quad 0 \leq y \leq 1\}\)

Define a relation R on A as follows: For all \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) in A,

\(\left(x_{1}, y_{1}\right) R\left(x_{2}, y_{2}\right) \Leftrightarrow\) \(\begin{array}{l}

\left(x_{1}, y_{1}\right)=\left(x_{2}, y_{2}\right) ; \text { or }\\

x_{1}=0 \quad \text { and } \quad x_{2}=1 \text { and } y_{1}=y_{2} ; \quad \text { or }\\

x_{1}=1 \quad \text { and } \quad x_{2}=0 \quad \text { and } \quad y_{1}=y_{2} ; \quad \text { or }\\

y_{1}=0 \quad \text { and } \quad y_{2}=1 \quad \text { and } \quad x_{1}=x_{2} ; \quad \text { or }\\

y_{1}=1 \quad \text { and } \quad y_{2}=0 \quad \text { and } \quad x_{1}=x_{2} \text {. }

\end{array}\)

In other words, all points along the top edge of the rectangle are related to the points along the bottom edge directly beneath them, and all points directly opposite each other along the left and right edges are related to each other. The points in the interior of the rectangle are not related to anything other than themselves. Then R is an equivalence relation on A. Imagine gluing together all the points that are in the same equivalence class. Describe the resulting figure.

Text Transcription:

A = {(x, y) in mathbf R times mathbf R mid 0 leq x leq 1 and 0 leq y leq 1}

(x_1, y_1)

(x_2, y_2)

(x_1, y_1) R (x_2, y_2) Leftrightarrow

(x_1, y_1) = (x_2, y_2) ; or

x_1 = 0 and x_2 = 1 and y_1 = y_2 ; or

x_1 = 1 and x_2 = 0 and y_1 = y_2 ; or

y_1 = 0 and y_2 = 1 and x_1 = x_2 ; or

y_1 = 1 and y_2 = 0 and x_1 = x_2.

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