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Chapter 2: Problem 14

Upper Bounds for Subsets of \(\mathbb{R} .\) Let \(A\) be a subset of the real numbers. A number \(b\) is called an upper bound for the set \(A\) provided that for each element \(x\) in \(A, x \leq b\) (a) Write this definition in symbolic form by completing the following: Let \(A\) be a subset of the real numbers. A number \(b\) is called an upper bound for the set \(A\) provided that \(\ldots . .\) (b) Give examples of three different upper bounds for the set \(A=\\{x \in \mathbb{R} \mid 1 \leq x \leq 3\\}\) (c) Does the set \(B=\\{x \in \mathbb{R} \mid x>0\\}\) have an upper bound? Explain. (d) Give examples of three different real numbers that are not upper bounds for the set \(A=\\{x \in \mathbb{R} \mid 1 \leq x \leq 3\\}\) (e) Complete the following in symbolic form: "Let \(A\) be a subset of \(\mathbb{R} .\) A number \(b\) is not an upper bound for the set \(A\) provided that \(\ldots . "\) (f) Without using the symbols for quantifiers, complete the following sentence: "Let \(A\) be a subset of \(\mathbb{R}\). A number \(b\) is not an upper bound for the set \(A\) provided that \(\ldots . "\) (g) Are your examples in Part (14d) consistent with your work in Part (14f)? Explain.

Short Answer

Expert verified
Part (a): Let \(A\) be a subset of the real numbers. A number \(b\) is called an upper bound for the set \(A\) provided that: \[\forall x \in A, \ x \leq b\] Part (b): For the given set \(A=\{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\), three different upper bounds are: \(b_1 = 3, \ b_2 = 4, \ \text{and} \ b_3 = 5\). Part (c): Set \(B = \{x \in \mathbb{R} \mid x > 0\}\) does not have an upper bound. Part (d): For the given set \(A=\{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\), three different real numbers that are not upper bounds are: \(b'_1 = 0, \ b'_2 = 1, \ \text{and} \ b'_3 = 2\). Part (e): A number \(b\) is not an upper bound for the set \(A\) provided that: \[\exists x \in A, \ x > b\] Part (f): "Let \(A\) be a subset of \(\mathbb{R}\). A number \(b\) is not an upper bound for the set \(A\) provided that there exists an element in A, which is greater than b." Part (g): The examples provided in Part (d) are consistent with the work in Part (f).

Step by step solution

01

Part (a) - Definition of an upper bound in symbolic form

Let \(A\) be a subset of the real numbers. A number \(b\) is called an upper bound for the set \(A\) provided that: \[\forall x \in A, \ x \leq b\]
02

Part (b) - Examples of three different upper bounds

For the given set \(A=\{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\), three different upper bounds are: \(b_1 = 3, \ b_2 = 4, \ \text{and} \ b_3 = 5\). These are all upper bounds, because for all \(x\) in set \(A\), it is true that \(x \leq b_i\) for \(i=1,2,3\).
03

Part (c) - The existence of an upper bound for set \(B\)

Set \(B = \{x \in \mathbb{R} \mid x > 0\}\) does not have an upper bound. Since \(x > 0\) for all \(x \in B\), there will never be a finite real number \(b\) such that \(x \leq b\) for all \(x \in B\).
04

Part (d) - Examples of three real numbers that are not upper bounds

For the given set \(A=\{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\), three different real numbers that are not upper bounds are: \(b'_1 = 0, \ b'_2 = 1, \ \text{and} \ b'_3 = 2\). These are not upper bounds because there are elements \(x \in A\) where \(x > b'_i\) for \(i=1,2,3\).
05

Part (e) - Symbolic form of "not an upper bound"

"Let \(A\) be a subset of \(\mathbb{R}\). A number \(b\) is not an upper bound for the set \(A\) provided that: \[\exists x \in A, \ x > b\]"
06

Part (f) - Completing the sentence without symbols for quantifiers

"Let \(A\) be a subset of \(\mathbb{R}\). A number \(b\) is not an upper bound for the set \(A\) provided that there exists an element in A, which is greater than b."
07

Part (g) - Consistency of examples with completed sentence

The examples provided in Part (d) are consistent with the work in Part (f). For all the numbers \(b'_i\) given in Part (d), there exists an element in the set \(A\) (e.g. \(x=3\) for \(b'_1, b'_2,\) and \(b'_3\)), which is greater than \(b'_i\).

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Key Concepts

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

Real Numbers
When we talk about real numbers, we refer to the set of numbers that include both rational and irrational numbers. The real numbers can be visualized as points on an infinitely long line, often called the number line. This continuous line includes all numbers such as fractions (like \(\frac{1}{2}\)) and irrational numbers (such as \(\pi\) or \(\sqrt{2}\)). Real numbers are fundamental in defining concepts like upper bounds in mathematics.
In our exercise, an upper bound is applied to a subset of real numbers. This concept utilizes the completeness property of real numbers, meaning every non-empty set bounded above at least has a least upper bound.
  • Examples of real numbers: \(2, -5, \frac{3}{4}, \sqrt{3}\)
  • Real numbers cover integers, whole numbers, and more.
Set Theory
Set theory is an elegant branch of mathematical logic that deals with sets, or collections of objects. In this context, when we refer to a subset of real numbers, we are talking about a specific portion of the infinite set of real numbers.
For example, the set \(A = \{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\) is a subset of real numbers from 1 to 3, inclusive. This subset represents just a slice of the real number line. Set theory allows us to manage and understand complex mathematical concepts by organizing objects into understandable groups.
  • Set notation is crucial for defining bounds.
  • A subset can be finite or infinite.
Symbolic Logic
Symbolic logic provides a way to express mathematical concepts with symbols. This form of logic lays out clear, concise statements, which are important when defining concepts like upper and non-upper bounds.
For instance, when saying "a number \(b\) is an upper bound for a set \(A\)", we can write it using symbolic logic as \(\forall x \in A, \ x \leq b\). This asserts that every element \(x\) in set \(A\) is less than or equal to \(b\). Similarly, if \(b\) is not an upper bound, this is expressed as \(\exists x \in A, \ x > b\), meaning there is at least one element in \(A\) greater than \(b\).
  • \(\forall\) stands for "for all".
  • \(\exists\) stands for "there exists".
Mathematical Examples
Mathematical examples help solidify abstract concepts by providing tangible instances. Let's take the set \(A = \{x \in \mathbb{R} \mid 1 \leq x \leq 3\}\) from the exercise. To illustrate what upper bounds are, consider the numbers \(3, 4,\) and \(5\). These act as upper bounds because each number in set \(A\) is less than or equal to them.
Conversely, the numbers \(0, 1,\) and \(2\) are not upper bounds. This is due to elements in the set (like \(x = 3\)) exceeding these values.
  • Upper bounds are larger than or equal to the maximum value in the set.
  • Non-upper bounds are smaller than some elements of the set.
These examples help us understand the utility of these concepts in problem-solving.

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