   Chapter 8.1, Problem 22ES ### 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
1 views

# Define relations R and S on R as follows: R = { ( x , y ) ∈ R × R|x 2 + y 2 = 4 }   and S= { ( x , y ) ∈ R × R|x=y } . Graph R,S, R ∪ S , and R ∩ S in the Cartesian plane.

To determine

Graph R, S, RS, and R nS in the Cartesian plane.

Explanation

Given information:

Define relations R and S onRas follows:

R = {(x, y)R xR | x2+ y2= 4} and

S = {(x, y)R xR | x = y}.

Calculation:

R = {(x, y)R xR | x2+y2=4 }

S = {(x, y)R xR | x = y}.

Graph R

R contains all ordered pairs (x, y) for which x2+y2=4.

However, x2+y2=4 represents the circle with radius 4=2 centered about the origin and thus the graph of R is the circle with radius 2 centered at the origin.

Graph S

S contains all ordered pairs (x, y) for which x = y, which is the straight line y = x.

Graph union:

The graph of the union RS contains al ordered pairs that belong to the region of R or belong to the region of S. thus the graph of the union then contains both the straight line y = x in the graph of S and the circle x2+y2=4 in the graph of R.

RS={(x,y)R×R|x2+y2=4 or x=y}

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