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Chapter 6: Problem 5

Let \(A\) and \(B\) be two nonempty sets. There are two projection functions with domain \(A \times B,\) the Cartesian product of \(A\) and \(B .\) One projection function will map an ordered pair to its first coordinate, and the other projection function will map the ordered pair to its second coordinate. So we define \(p_{1}: A \times B \rightarrow A\) by \(p_{1}(a, b)=a\) for every \((a, b) \in A \times B ;\) and \(p_{2}: A \times B \rightarrow B\) by \(p_{2}(a, b)=b\) for every \((a, b) \in A \times B\) Let \(A=\\{1,2\\}\) and let \(B=\\{x, y, z\\}\) (a) Determine the outputs for all possible inputs for the projection function \(p_{1}: A \times B \rightarrow A\) (b) Determine the outputs for all possible inputs for the projection function \(p_{2}: A \times B \rightarrow B\) (c) What is the range of these projection functions? (d) Is the following statement true or false? Explain. For all \((m, n),(u, v) \in A \times B,\) if \((m, n) \neq(u, v),\) then \(p_{1}(m, n) \neq p_{1}(u, v)\)

Short Answer

Expert verified
The Cartesian product A x B is {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)}. For projection functions p1 and p2, we have: p1: (1,x)鈫�1, (1,y)鈫�1, (1,z)鈫�1, (2,x)鈫�2, (2,y)鈫�2, (2,z)鈫�2, p2: (1,x)鈫抶, (1,y)鈫抷, (1,z)鈫抸, (2,x)鈫抶, (2,y)鈫抷, (2,z)鈫抸. The ranges of p1 and p2 are p1: {1, 2}, p2: {x, y, z}. The statement "For all (m, n), (u, v) in A x B, if (m, n) 鈮� (u, v), then p1(m, n) 鈮� p1(u, v)" is false because, for example, (1,x) 鈮� (1,y) but p1(1,x) = p1(1,y) = 1.

Step by step solution

01

Determine the Cartesian Product A x B

Given A = {1, 2} and B = {x, y, z}, we can find the Cartesian product A x B by creating ordered pairs with each element from A as the first coordinate and each element from B as the second coordinate. Therefore, A x B = {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)}.
02

Find the outputs of p1 and p2

Now, we'll find the output for the projection functions p1 and p2 for each element in A x B. Using the definitions of p1 and p2: p1(a, b) = a for every (a, b) in A x B, and p2(a, b) = b for every (a, b) in A x B, For the input elements from A x B: p1(1,x) = 1, p2(1,x) = x, p1(1,y) = 1, p2(1,y) = y, p1(1,z) = 1, p2(1,z) = z, p1(2,x) = 2, p2(2,x) = x, p1(2,y) = 2, p2(2,y) = y, p1(2,z) = 2, p2(2,z) = z.
03

Identify the ranges of p1 and p2

The range of a function consists of its output values when applied to its domain. In this problem, the domain of p1 and p2 is A x B. Since p1 takes the first coordinate value, its range is set A. Similarly, p2 takes the second coordinate value, so its range is set B. Thus, the range of p1 is A = {1, 2}, and the range of p2 is B = {x, y, z}.
04

Evaluate the given statement

The statement to evaluate says: "For all (m, n), (u, v) in A x B, if (m, n) 鈮� (u, v), then p1(m, n) 鈮� p1(u, v)." This statement is false. We can provide a counterexample: let (m, n) = (1,x) and (u, v) = (1,y). We have (m, n) 鈮� (u, v) because x 鈮� y, but p1(m, n) = p1(1,x) = 1 and p1(u, v) = p1(1,y) = 1, hence p1(m, n) = p1(u, v). This counterexample disproves the statement, demonstrating that it is false.

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Key Concepts

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

Cartesian Product
In mathematics, the Cartesian product is a fundamental concept used to create ordered pairs from two sets. Given two non-empty sets, say set \( A \) and set \( B \), the Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element comes from \( A \) and the second comes from \( B \). This procedure essentially pairs every element of \( A \) with every element of \( B \). For example, if \( A = \{1, 2\} \) and \( B = \{x, y, z\} \), the Cartesian product \( A \times B \) is:
  • (1, x)
  • (1, y)
  • (1, z)
  • (2, x)
  • (2, y)
  • (2, z)
Understanding Cartesian products is essential when dealing with functions and relations because they provide the domain from which we can project functions between sets.
Range of a Function
The range of a function is the set of all possible outputs that the function can produce when applied to its domain. It is essentially the collection of values that results from substituting every possible input value. For the projection functions \( p_1: A \times B \rightarrow A \) and \( p_2: A \times B \rightarrow B \), the output range differs based on the components being projected.
  • The function \( p_1 \) takes the first component from each ordered pair. Hence, its range is simply the set \( A \). For the problem at hand, this is \( \{1, 2\} \).
  • The function \( p_2 \) takes the second component from each ordered pair. Thus, its range is the set \( B \), which in our example is \( \{x, y, z\} \).
This understanding allows you to easily determine the possible outcomes of a function, aiding in mapping the projected elements accurately.
Counterexample
A counterexample is a specific case that disproves a general statement or hypothesis. In mathematics, it's an essential tool to demonstrate that a universally quantified statement is false. The process involves finding even one instance where the statement does not hold. In our exercise, a statement was given about the behavior of ordered pairs under the projection function \( p_1: A \times B \rightarrow A \). The statement claimed: "For all \((m, n), (u, v) \in A \times B\), if \((m, n) eq (u, v)\), then \( p_1(m, n) eq p_1(u, v)\)." By identifying the pair \((m, n) = (1, x)\) and \((u, v) = (1, y)\), where clearly \((1, x) eq (1, y)\) but \(p_1(1, x) = p_1(1, y) = 1\), we provided a counterexample. This example directly contradicts the statement, proving it is false. Learning how to find and use counterexamples can deepen understanding and strengthens one's problem-solving skills.
Sets and Relations
Sets and relations form the backbone of many mathematical concepts, providing the language and structure for discussing collections of objects and the connections between them. - A **set** is simply a collection of distinct objects, known as elements. Sets are usually denoted with curly braces. For example, \( \{1, 2, 3\} \) is a set containing the numbers 1, 2, and 3.- A **relation** is any association between elements from one set to another. In formal terms, a relation \( R \) from set \( A \) to set \( B \) is a subset of the Cartesian product \( A \times B \). It includes only the ordered pairs \((a, b)\) where \( a \) is related to \( b \) in some way defined by \( R \).Understanding sets simplifies the exploration of relations, because it allows us to consider how elements compare or link across diverse sets. These concepts are underpinning many areas of mathematics, from algebra to computer science.

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Most popular questions from this chapter

Let \(A\) be a nonempty set and let \(f: A \rightarrow A\). For each \(n \in \mathbb{N}\), define a function \(f^{n}: A \rightarrow A\) recursively as follows: \(f^{1}=f\) and for each \(n \in \mathbb{N}\) \(f^{n+1}=f \circ f^{n} .\) For example, \(f^{2}=f \circ f^{1}=f \circ f\) and \(f^{3}=f \circ f^{2}=\) \(f \circ(f \circ f)\) (a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x)=x+1\) for each \(x \in \mathbb{R}\). For each \(n \in \mathbb{N}\) and for each \(x \in \mathbb{R}\), determine a formula for \(f^{n}(x)\) and use induction to prove that your formula is correct. (b) Let \(a, b \in \mathbb{R}\) and let \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x)=a x+b\) for each \(x \in \mathbb{R}\). For each \(n \in \mathbb{N}\) and for each \(x \in \mathbb{R},\) determine a formula for \(f^{n}(x)\) and use induction to prove that your formula is correct. (c) Now let \(A\) be a nonempty set and let \(f: A \rightarrow A\). Use induction to prove that for each \(n \in \mathbb{N}, f^{n+1}=f^{n} \circ f .\) (Note: You will need to use the result in Exercise (5).)

Recall that a real function is a function whose domain and codomain are subsets of the real numbers \(\mathbb{R}\). (See page 288.) Most of the functions used in calculus are real functions. Quite often, a real function is given by a formula or a graph with no specific reference to the domain or the codomain. In these cases, the usual convention is to assume that the domain of the real function \(f\) is the set of all real numbers \(x\) for which \(f(x)\) is a real number, and that the codomain is \(\mathbb{R}\). For example, if we define the (real) function \(f\) by $$ f(x)=\frac{x}{x-2} $$ we would be assuming that the domain is the set of all real numbers that are not equal to 2 and that the codomain is \(\mathbb{R}\). Determine the domain and range of each of the following real functions. It might help to use a graphing calculator to plot a graph of the function. (a) The function \(k\) defined by \(k(x)=\sqrt{x-3}\) (b) The function \(F\) defined by \(F(x)=\ln (2 x-1)\) (c) The function \(f\) defined by \(f(x)=3 \sin (2 x)\) (d) The function \(g\) defined by \(g(x)=\frac{4}{x^{2}-4}\) (e) The function \(G\) defined by \(G(x)=4 \cos (\pi x)+8\)

Let \(g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) be defined by \(g(m, n)=(2 m, m-n)\). (a) Calculate \(g(3,5)\) and \(g(-1,4)\) (b) Determine all the preimages of (0,0) . That is, find all \((m, n) \in \mathbb{Z} \times \mathbb{Z}\) such that \(g(m, n)=(0,0)\) (c) Determine the set of all the preimages of (8,-3) . (d) Determine the set of all the preimages of (1,1) . (e) Is the following proposition true or false? Justify your conclusion. For each \((s, t) \in \mathbb{Z} \times \mathbb{Z},\) there exists an \((m, n) \in \mathbb{Z} \times \mathbb{Z}\) such that \(g(m, n)=(s, t)\)

(a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=x^{2} .\) Explain why the inverse of \(f\) is not a function. (b) Let \(\mathbb{R}^{*}=\\{t \in \mathbb{R} \mid t \geq 0\\} .\) Define \(g: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}\) by \(g(x)=x^{2} .\) Ex- plain why this squaring function (with a restricted domain and codomain) is a bijection. (c) Explain how to define the square root function as the inverse of the function in Exercise (8b). (d) True or false: \((\sqrt{x})^{2}=x\) for all \(x \in \mathbb{R}\) such that \(x \geq 0\) (e) True or false: \(\sqrt{x^{2}}=x\) for all \(x \in \mathbb{R}\)

For each of the following, give an example of functions \(f: A \rightarrow B\) and \(g: B \rightarrow C\) that satisfy the stated conditions, or explain why no such example exists. "(a) The function \(f\) is a surjection, but the function \(g \circ f\) is not a surjection. (b) The function \(f\) is an injection, but the function \(g \circ f\) is not an injection. (c) The function \(g\) is a surjection, but the function \(g \circ f\) is not a surjection. (d) The function \(g\) is an injection, but the function \(g \circ f\) is not an injection. (e) The function \(f\) is not a surjection, but the function \(g \circ f\) is a surjection. (f) The function \(f\) is not an injection, but the function \(g \circ f\) is an injection. (g) The function \(g\) is not a surjection, but the function \(g \circ f\) is a surjection. (h) The function \(g\) is not an injection, but the function \(g \circ f\) is an injection.
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