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Chapter 9: Problem 7

Define a binary relation \(R\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: For all \((x, y) \in \mathbf{R} \times \mathbf{R}, \quad x R y \quad \Leftrightarrow \quad y=x^{2} .\) a. Is \((2,4) \in R ?\) Is \((4,2) \in R ?\) Is \((-3) R 9 ?\) Is \(9 R(-3) ?\) b. Draw the graph of \(R\) in the Cartesian plane.

Short Answer

Expert verified
a. (2,4) 鈭� R, (4,2) 鈭� R, (-3)R9, 9R(-3) 鈭� R b. The graph of R is a parabola with vertex at (0,0) opening upwards, and axis of symmetry x=0.

Step by step solution

01

Checking if (2,4) is in R

Since the relation is defined as \(xRy \Leftrightarrow y=x^2\), let's substitute \(x=2\) and \(y=4\). That is, \(4=2^2\). As this is true, \((2,4) \in R\).
02

Checking if (4,2) is in R

Now, let's substitute \(x=4\) and \(y=2\) in the relation definition: \(2=4^2\). This statement is false, so \((4,2) \notin R\).
03

Checking if (-3)R9

For \(x=-3\) and \(y=9\), let's see if \(9=(-3)^2\). This statement is true, so \((-3)R9\).
04

Checking if 9R(-3)

Lastly, let's substitute \(x=9\) and \(y=-3\) in the relation definition: \(-3=9^2\). This statement is false, so \(9R(-3)\) is not in the relation. b. Drawing the graph of R:
05

Setting up the graph

To draw the graph of this relation, we'll use the fact that \(y=x^2\). Since it is a quadratic function, its graph will be a parabola.
06

Identifying key features

The vertex of the parabola is at point (0,0), and it opens upwards. The axis of symmetry is the y-axis or x=0.
07

Sketching the graph

To sketch the graph, first plot the vertex (0,0) and a few other points such as (1,1), (-1,1), (2,4), and (-2,4). Next, draw a smooth curve through these points to create the parabola, making sure it is symmetrical about the y-axis (x=0). Now we have analyzed the relation R and drawn its graph. To summarize: a. (2,4) is in R, (4,2) is not in R, (-3)R9, and 9R(-3) is not b. The graph of R is a parabola with vertex at (0,0) opening upwards, and axis of symmetry x=0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian Plane Graphing
Graphing on the Cartesian plane is a fundamental skill in mathematics that involves plotting points, lines, and curves to visually represent mathematical relationships. The plane is formed by two perpendicular number lines: the horizontal axis (x-axis) and the vertical axis (y-axis). Each point on this plane can be defined by an ordered pair (x, y) representing its coordinates.

Cartesian plane graphing provides a powerful way to visualize a binary relation such as the one defined in the exercise, which matches each x-value with a y-value following a specific rule, here, y = x^2. Placing points on the graph according to this rule and connecting them with a smooth line or curve helps in understanding the nature of the relation.
  • To graph a relation, first identify a set of ordered pairs that satisfy the relation.
  • Next, plot these pairs on the plane.
  • Finally, connect the plotted points appropriately to reveal the pattern or shape of the relation.
Graphing is not merely about plotting; it also includes understanding the shape and the properties of the graph, such as symmetry, intercepts, and in the case of functions, understanding their domain and range.
Quadratic Function
A quadratic function can be recognized in its standard form: f(x) = ax^2 + bx + c, where a, b, and c are constants and 'a' is not zero. This particular function creates a parabola when graphed on the Cartesian plane.

When a quadratic function is graphed, the parabola can open upwards or downwards, which is determined by the sign of coefficient 'a'. If 'a' is positive, the parabola opens upwards like a regular 'U,' and if 'a' is negative, it opens downwards like an inverted 'U.'

Features of a Parabola

  • Vertex: The highest or lowest point of a parabola, located at the axis of symmetry.
  • Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves.
  • Intercepts: Points where the parabola crosses the x-axis (x-intercepts) and y-axis (y-intercept).
A key property of a quadratic function is that for every x-value, there is only one corresponding y-value, which establishes it as a function and specifies its relation properties.
Relation Properties
Understanding relation properties is essential in deciphering and graphing binary relations. In mathematics, a binary relation R between two sets A and B is a collection of ordered pairs (a, b) where a is in A and b is in B.

A relation can have several properties which define its behavior and constraints:

Reflexivity, Symmetry, and Transitivity

  • Reflexive: If every element is related to itself, for example, (x,x).
  • Symmetric: If for every (a,b) in the relation, (b,a) is also in the relation.
  • Transitive: If whenever (a,b) and (b,c) are in the relation, then (a,c) must also be.
In the context of the exercise, the relation is described by the rule y = x^2. This defines a particular kind of relation known as a function. A function is a special type of relation with the property that every x-value is associated with exactly one y-value. This property is pivotal because it dictates the way the relation is represented on a graph and further defines the behavior of mathematical models described by such relations.

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Most popular questions from this chapter

Let \(f\) be a real-valued function of a real varaible. Show that if \(f\) is increasing on a set \(S\) and if \(M\) is any negative real number, then \(M f\) is decreasing on \(S\).

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Let \(k: \mathbf{R} \rightarrow \mathbf{R}\) be the function defined by the formula \(k(x)=\) \((x-1) / x\) for all real numbers \(x \neq 0\). a. Show that \(k\) is increasing for all real numbers \(x>0\). b. Is \(k\) increasing or decreasing for \(x<0\) ? Prove your answer.

Exercises 4-5 explore the fact that for relatively small values of \(n\), algorithms with larger orders can be more efficient than algorithms with smaller orders. Suppose that when run with an input of size \(n\), algorithm \(A\) requires \(10^{6} n^{2}\) operations and algorithm \(B\) requires \(n^{3}\) operations. a. What are orders for algorithms \(A\) and \(B\) from among the set of power functions? b. For what values of \(n\) is algorithm \(A\) more efficient than algorithm \(B\) ? c. For what values of \(n\) is algorithm \(B\) at least 100 times more efficient than algorithm \(A ?\)

Draw the graphs of the power functions \(p_{1 / 3}\) and \(p_{1 / 4}\) on the same set of axes. When \(01\), which is greater: \(x^{1 / 3}\) or \(x^{1 / 4}\) ?
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