   Chapter 8.3, Problem 33ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Let A be the set of points in the rectangle withxandy coordinates between 0 and 1. That is.   A = { ( x , y ) ∈ R × R | 0 ≤ x ≤ 1   and   0 ≤ y ≤ 1 } . Define a relation R on A as follows: For all ( x 1 ,   y 1 ) and ( x 2 , y 2 ) in A, 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 alone 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 plants that are in the same equivalence class. Describe the resulting figure

To determine

Describe the resulting figure based on the given information.

Explanation

Given information:

Define a relation R on A as follows: For all ( x1, y1 ) and ( x2, y2 ) in A ,

( x1, y1 ) R ( x2, y2 ) ⇔ ( x1, y1 ) = ( x2, y2 ); or

x1 =0 and x2 =1 and y1 = y2 ; or

x1 =1 and x2 =0 and y1 = y2 ; or

y1 =0 and y2 =1 and x1 = x2 ; or

y1 =1 and y2 =0 and x1 = x2.

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.

Calculation:

A={(x,y)R×R|0x1 and 0y1}R={(( x 1 , y 1 ),( x 2 , y 2 ))|( x 1 , y 1 )=( x 2 , y 2 ) or                                    (x1=0  and x2

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