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Chapter 9: Problem 32

Define binary relations \(R\) and \(S\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: $$ \begin{aligned} &R=\left\\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x^{2}+y^{2}=4\right\\} \quad \text { and } \\ &S=\\{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x=y\\} . \end{aligned} $$ Graph \(R, S, R \cup S\), and \(R \cap S\) in the Cartesian plane.

Short Answer

Expert verified
In summary, relation R represents a circle with radius 2 centered at the origin, while relation S represents a straight line with a 45-degree angle from the horizontal axis, passing through the origin. Their union consists of all points that belong to either R or S or both, which is a graph containing both the circle and the straight line. The intersection of R and S consists of two points, \((\sqrt{2},\sqrt{2})\) and \((- \sqrt{2}, -\sqrt{2})\), where the circle and the line meet.

Step by step solution

01

Identify and understand each relation

Let's examine the two relations. 1. \(R = \{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x^{2}+y^{2}=4\}\) This is a relation defined as the set of all ordered pairs (x,y) of real numbers that satisfy the equation \(x^2 + y^2 = 4\). This represents a circle with radius 2 and centered at the origin (0,0). 2. \(S = \{(x, y) \in \mathbf{R} \times \mathbf{R} \mid x=y\}\) This is a relation defined as the set of all ordered pairs (x,y) of real numbers that satisfy the equation \(x = y\). This represents a straight line with a 45-degree angle from the horizontal axis, passing through the origin (0,0).
02

Graph the relations R and S individually

Now let's graph the relations R and S separately: 1. Graph relation R: Draw a circle with radius 2 centered at the origin (0,0). 2. Graph relation S: Draw a straight line with a 45-degree angle from the horizontal axis, passing through the origin (0,0).
03

Graph the union of R and S

To graph the union of relations R and S, simply combine both relations' graphs. This means drawing the circle from R and the straight line from S on the same Cartesian plane. The relation \(R \cup S\) consists of all points that belong to either R or S or both.
04

Graph the intersection of R and S

To graph the intersection of relations R and S, we need to find the points at which both relations intersect, i.e., the points that are present in both R and S. Since relation R equation is \(x^2 + y^2 = 4\) and relation S equation is \(x = y\), we can substitute x by y in relation R equation to find the intersection points: \((y)^2 + y^2 = 4\) \(2y^2 = 4\) \(y^2 = 2\) \(y = \pm \sqrt{2}\) When \(y = \sqrt{2}\), \(x = \sqrt{2}\), and when \(y = -\sqrt{2}\), \(x = -\sqrt{2}\). So, the intersection points of R and S are \((\sqrt{2},\sqrt{2})\) and \((- \sqrt{2}, -\sqrt{2})\). Now, graph the intersection points of R and S on the Cartesian plane. The relation \(R \cap S\) consists of only these two points.

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Most popular questions from this chapter

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