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Chapter 1: Problem 4

Determine whether each statement is true or false. If it is false, explain why. The intersection of the sets \((-\infty, 7]\) and \([7, \infty)\) is \\{7\\} .

Short Answer

Expert verified
The statement is true. The intersection is \{7\}.

Step by step solution

01

- Understand the given sets

Identify the elements in the sets \((-\infty, 7]\) and \([7, \infty)\). The set \((-\infty, 7]\) includes all real numbers less than or equal to 7. The set \([7, \infty)\) includes all real numbers greater than or equal to 7.
02

- Find the intersection

The intersection of two sets is the set of elements that are common to both sets. For the given sets \((-\infty, 7]\) and \([7, \infty)\), the only common element is 7.
03

- Conclusion

Since the only common element in both sets \((-\infty, 7]\) and \([7, \infty)\) is 7, the intersection of these sets is indeed \{7\}. Therefore, the statement is true.

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Key Concepts

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

Real Numbers
Real numbers represent all the possible numbers along the number line. This includes integers, fractions, and irrational numbers like \(\frac{\text{pi}}{2}\) or \( \sqrt{2} \). Real numbers can be both positive and negative, including zero. Because real numbers cover both rational and irrational numbers, they are a broad and inclusive set.

Understanding real numbers is essential because they are the foundation of many mathematical concepts, including the sets involved in this exercise. When dealing with intervals like \( -\infty, 7] \) and \([7, \infty)\), we need to be aware that all numbers within these intervals are real numbers, including decimals and irrational numbers.
Interval Notation
Interval notation is a mathematical notation used to describe a range of numbers within the real number system. Instead of listing every single number, we use intervals to succinctly indicate which numbers are included.

In this exercise, the intervals are \( -\infty, 7] \) and \([7, \infty)\). Here's what each part means:
  • The interval \( -\infty, 7] \) includes all real numbers less than or equal to 7. The square bracket \(]\) means that 7 is included. The symbol \( -\infty \) indicates that the interval extends infinitely to the negative side.
  • The interval \([7, \infty)\) includes all real numbers greater than or equal to 7. The square bracket \([\) means that 7 is included. The symbol \(\infty \) indicates that the interval extends infinitely to the positive side.
Knowing how to read and write interval notation helps you easily understand the range of numbers you are working with, which is critical in set theory.
Set Theory
Set theory is a fundamental branch of mathematical logic that deals with sets, which are collections of objects. Here are some basic concepts:
  • A set is a collection of distinct objects, considered as an object in its own right.
  • The elements of a set are the objects that belong to it.
  • The intersection of two sets is the set of elements that are common to both sets.
In this exercise, we are asked to find the intersection of two sets: \( -\infty, 7] \) and \([7, \infty)\).

When we find the intersection of these two sets, we identify the elements that both sets share. In this case, the only common element is 7. So, the intersection is {7}.

Understanding how to find the intersection of sets is a key part of set theory, and it's useful for solving many real-world problems that involve grouping and overlapping categories.

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Most popular questions from this chapter

Between 2000 and \(2016,\) the estimated population of metro New Orleans, Louisiana, declined from 1,337,726 to \(1,268,883 .\) What was the percent decrease to the nearest tenth?

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information. Work each of the following. (a) Write an absolute value inequality that describes the height of a building that is not within \(95 \mathrm{ft}\) of the average. (b) Solve the inequality from part (a). (c) Use the result of part (b) to list the buildings that are not within \(95 \mathrm{ft}\) of the average. (d) Confirm that the answer to part (c) makes sense by comparing it with the answer to Exercise 131 .

Solve each equation or inequality. $$ |7 x+3| \leq 0 $$

Solve each percent problem. Why is it impossible to mix candy worth \(\$ 4\) per \(1 \mathrm{~b}\) and candy worth \(\$ 5\) per Ib to obtain a final mixture worth \(\$ 6\) per \(1 \mathrm{~b} ?\)

A jar of 20 coins contains 4 pennies. What percent of the coins are pennies?
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