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Chapter 5: Problem 14

Let \(A=B=\mathbf{R}\). Determine \(\pi_{A}(D)\) and \(\pi_{B}(D)\) for each of the following sets \(D \subseteq A \times B\). g) \(D=\left\\{(x, y) \mid x=y^{2}\right\\}\) b) \(D=\\{(x, y) \mid y=\sin x\\}\) c) \(D=\left\\{(x, y) \mid x^{2}+y^{2}=1\right\\}\)

Short Answer

Expert verified
\(\pi_{A}(D)=\pi_{B}(D)=[0, +\infty)\) for subset g), \(\pi_{A}(D)=\mathbf{R}\) and \(\pi_{B}(D)=[-1, 1]\) for subset b), and \(\pi_{A}(D)=\pi_{B}(D)=[-1, 1]\) for subset c).

Step by step solution

01

Identify and project subset g)

Subset g) is defined as \(D=\left\{(x, y) \mid x=y^{2}\right\}\). From this, we can deduce that the projection of \(D\) onto \(A\) will include all \(x\) such that \(x\) is equal to the square of some real number, i.e., \(\pi_{A}(D) = [0, +\infty)\). And the projection of \(D\) onto \(B\), \(\pi_{B}(D)\), will include all \(y\) such that \(y\) is the square root of some \(x\). In the domain of real numbers, this means \(y\) can only be non-negative, i.e., \(\pi_{B}(D) = [0, +\infty)\).
02

Identify and project subset b)

Subset b) is defined as \(D=\{(x, y) \mid y=\sin x\}\). The projection of \(D\) onto \(A\), \(\pi_{A}(D)\), is the set of all \(x\) values. As \(x\) is a real number, \(\pi_{A}(D) = \mathbf{R}\). Meanwhile, the projection of \(D\) onto \(B\), \(\pi_{B}(D)\), is the set of all \(y\) such that \(y\) is equal to sine of some value. The range of \(\sin x\) is [-1, 1], therefore, \(\pi_{B}(D) = [-1, 1]\).
03

Identify and project subset c)

Subset c) is defined as \(D=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}\). The \(D\) defines a unit circle. The projection onto \(A\), \(\pi_{A}(D)\), will include all \(x\) that lie on the unit circle, which is the interval [-1, 1]. Similarly, the projection onto \(B\), \(\pi_{B}(D)\), will include all \(y\) that lie on the unit circle, which is also the interval [-1, 1].

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Key Concepts

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

Projections
Projections are an essential concept in set theory that help with understanding the behavior of sets when mapped onto axes or spaces. When we have a set of ordered pairs, like a subset of a Cartesian plane, projections allow us to focus on one dimension at a time.
Consider a set \(D = \{(x, y) \mid x=f(y) \}\). We can "project" this set onto the \(x\)-axis, known as the projection onto \(A\), denoted by \(\pi_{A}(D)\). This results in a set of \(x\)-values that can be obtained when the equation \(x = f(y)\) is satisfied. For example, if \(f(y) = y^2\), then \(x = y^2\) makes \(x\) belong specifically to \([0, \infty)\).
  • Projections onto \(B\) are similar and focus on the \(y\)-dimension, using the form \(\pi_{B}(D)\), where you derive the set of \(y\)-values from the equation \(y = g(x)\).
  • Through mapping in the real number plane, projections can distill complex relationships into clear, separate dimensions.
This technique can simplify analysis and visualization, especially when dealing with infinite sets or complex equations.
Real Numbers
Real numbers are the foundation of both algebraic functions and geometric representations. They encompass all numbers you learn about in various math courses, including the rationals like fractions and whole numbers, and irrationals like \(\pi\) and \(\sqrt{2}\).
This domain supports the full spectrum of number operations and analyses. When dealing with projections, real numbers provide valuable insights and limit our solutions to feasible, comprehensible values.
  • As seen in the given solutions, the projection of an equation into a real number plane assists in translating real equations (like \(y = \sin(x)\) or \(x = y^2\)) into observable, measurable outcomes in the \(x\)- or \(y\)-directions.
  • They allow for a complete, uninterrupted analysis, whether you're looking for solutions from \([-\infty, \infty]\), or are limited to specific intervals like \([0, \infty)\) due to function definitions.
Unit Circle
The unit circle is a central concept in trigonometry and geometry. It refers to the circle on the Cartesian plane with a radius of one, centered at the origin, defined by the equation \(x^2 + y^2 = 1\). This circle offers a splendid visual manifestation of trigonometric functions and their relationships.
  • Every point on the unit circle has coordinates that satisfy \(x^2 + y^2 = 1\), offering immediate insights into angles and sine-cosine relationships.
  • Traditionally, in projected sets like \(D = \{(x, y) \mid x^2 + y^2 = 1\}\), projections onto axes reflect identifiable intervals, \([-1, 1]\), revealing how \(x\) and \(y\) encompass values from negative one to positive one inclusively.
Understanding the unit circle aids significantly in visualizing periodic functions and oscillatory behaviors, building a bridge between algebraic representations and geometric interpretations.

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Most popular questions from this chapter

Let \(A_{i}, 1 \leq i \leq 5\), be the domains for a table \(D \subseteq \times_{i=1}^{5} A_{i}\), where \(A_{1}=\\{\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}, \mathrm{Z}\\}\) (used as code names for different cereals in a test), and \(A_{2}=A_{3}=A_{4}=A_{5}=\mathbf{Z}^{+} .\)The table \(D\) is given as Table \(5.7 .\) Table \(5.7\) \begin{tabular}{|c|c|c|c|c|} \hline Code Name of Cereal & Grams of Sugar per 1-0z Serving & \% of RDA \(^{\text {t }}\) of Vitamin A per 1-0z Serving & \% of RDA of Vitamin C per 1-0z Serving & \% of RDA of Protein per 1-0z Serving \\ \hline U & 1 & 25 & 25 & 6 \\ V & 7 & 25 & 2 & 4 \\ W & 12 & 25 & 2 & 4 \\ X & 0 & 60 & 40 & 20 \\ Y & 3 & 25 & 40 & 10 \\ Z & 2 & 25 & 40 & 10 \\ \hline \end{tabular} \({ }^{a} \mathrm{RDA}=\) rccommended daily allowance a) What is the degree of the table? b) Find the projection of \(D\) on \(A_{3} \times A_{4} \times A_{5}\). c) A domain of a table is called a primary key for the table if its value uniquely identifies each list of \(D\). Determine the primary key(s) for this table.

a) Let \(S \subset \mathbf{Z}^{+}\). What is the smallest value for \(|S|\) that guarantees the existence of two elements \(x, y \in S\) where \(x\) and \(y\) have the same remainder upon division by 1000 ? b) What is the smallest value of \(n\) such that whenever \(S \subseteq \mathbf{Z}^{+}\)and \(|S|=n\), then there exist three elements \(x, y, z \in S\) where all three have the same remainder upon division by 1000 . c) Write a statement that generalizes the results of parts \((\mathrm{a})\) and \((\mathrm{b})\) and Example \(5.43\).

a) For \(A=\\{1,2,3, \ldots, 7\\}\), how many bijective functions \(f: A \rightarrow A\) satisfy \(f(1) \neq 1\) ? b) Answer part (a) for \(A=\left\\{x \mid x \in \mathbf{Z}^{+}, 1 \leq x \leq n\right\\}\).

Let \(A=\\{1,2,3,4,5\\}\) and \(B=\\{t, u, v, w, x, y, z\\}\). (a) If a function \(f: A \rightarrow B\) is randomly generated, what is the probability that it is one-to-one? (b) Write a computer program (or develop an algorithm) to generate random functions \(f: A \rightarrow B\) and have the program print out how many functions it generates until it generates one that is one-to-one.

Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be defined by \(f(x)=x^{2}\). For each of the following subsets \(B\) of \(\mathbf{R}\), find \(f^{-1}(B)\). a) \(B=\\{0,1\\}\) b) \(B=\\{-1,0,1\\}\) c) \(B=[0,1]\) d) \(B=[0,1)\) e) \(B=[-1,1]\) f) \(B=[0,4]\) g) \(B=[0,1] \cup[4,9]\) h) \(B=(0,1] \cup(4,9)\) Determine three infinite subsets \(B\) of \(\mathbf{R}\) for which \(f^{-1}(B)=\emptyset\).
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