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Chapter 5: Problem 16

Rewrite the set using set-builder notation. $$F=\\{-1,0,1,2,3,4,5\\}$$

Short Answer

Expert verified
Set-builder notation: \( F = \{ n \in \mathbb{Z} \mid -1 \leq n \leq 5 \} \).

Step by step solution

01

Understand the Set

The set \( F \) is given as \( \{-1, 0, 1, 2, 3, 4, 5\} \). We need to identify a rule or property that all elements of this set share, which will help us express it in set-builder notation.
02

Identify the Mathematical Rule

Upon observing the elements of the set \( F \), we notice the elements are consecutive integers starting from \(-1\) and ending at \(5\). Thus, we can describe these elements using the property \( n \) where \( n \) is an integer such that \(-1 \leq n \leq 5\).
03

Write in Set-builder Notation

Using the identified rule, we can express the set \( F \) in set-builder notation as: \( F = \{ n \in \mathbb{Z} \mid -1 \leq n \leq 5 \} \). This notation reads as the set of all integers \( n \) such that \( n \) is greater than or equal to \(-1\) and less than or equal to \(5\).

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Key Concepts

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

Integers
Integers are a fundamental part of mathematics. They include all whole numbers and their negatives. Simply put, integers are like the counting numbers you learned as a child, but without any fractions or decimals. They can be positive, negative, or zero.
Here are some examples:
  • -3, -2, -1: These are negative integers.
  • 0: This is neither positive nor negative, and it's important because it serves as the bridge between positive and negative numbers.
  • 1, 2, 3: These are positive integers.
In the context of the original exercise, integers help in identifying the complete set of numbers we are dealing with when rewriting sets, such as set \( F = \{-1, 0, 1, 2, 3, 4, 5\} \). Each element in this set is an integer.
Consecutive Numbers
Consecutive numbers are numbers that follow each other in order, without any gaps. This means each number is one unit larger than the previous one. For example, 1, 2, and 3 are consecutive because they are in a direct sequence.
To elaborate further, consider:
  • The numbers -1 and 0, 0 and 1, all the way up to 4 and 5, are consecutive in the set \( F \) you are working with.
When expressing this set in set-builder notation, recognizing that the numbers are consecutive helps define the range of the integers in the set. In set-builder notation, we express this through the condition \(-1 \leq n \leq 5\) which linearly maps out the start and the end of this sequence without skips or omissions.
Mathematical Sets
A mathematical set is simply a collection of distinct objects or numbers, considered as an object in its own right. Sets are fundamental in all branches of mathematics.
Here's a deeper dive into sets:
  • Sets are usually represented with curly braces \( \{ \} \), and the list of elements inside them. The order of elements does not matter, nor does repeating elements.
  • The original set \( F = \{-1, 0, 1, 2, 3, 4, 5\} \) can be described using set-builder notation which includes an expression defining the properties that characterize the elements of the set.
  • In set-builder notation, \( F = \{ n \in \mathbb{Z} \mid -1 \leq n \leq 5 \} \) indicates all integers \( n \) such that \( n \) is within this specific range and belongs to the set of integers \( \mathbb{Z} \).
Understanding sets and their notation is crucial for clarifying which elements belong to a particular collection or grouping and how they relate to each other through specific conditions or rules.

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Most popular questions from this chapter

Consider a March 2015 Gallup poll in which 1025 randomly chosen American adults were asked if they believe that more emphasis should be placed on traditional sources of energy, such as fossil fuels and nuclear energy, or on alternative energy sources, such as wind and solar energy. The respondents were also asked to classify their party affiliation. The results are shown in the contingency table. $$\begin{array}{lccc}\hline & \begin{array}{c}\text { More Traditional } \\\\\text { Sources }\end{array} & \begin{array}{c}\text { More Alternative } \\\\\text { Sources }\end{array} & \text { Total } \\\\\hline \text { Democrat } & 107 & 498 & 605 \\\\\text { Republican } & 252 & 168 & 420 \\ \text { Total } & 359 & 666 & 1025 \\\\\hline\end{array}$$ Answer the following questions. Round to the nearest whole percentage as necessary. What percentage of those in the survey were either Democrats or said that more emphasis should be placed on traditional energy sources?

A contingency table is given without row and column totals. Determine the cardinality in each case. $$\begin{array}{lcc}\hline & \text { Set A } & \text { Set B } \\\\\hline \text { Set C } & 26 & 15 \\\\\text { Set D } & 11 & 28 \\\\\hline\end{array}$$ $$n(B \cap C)$$

Consider a survey in Pet Palace magazine in which single adults were asked if they own a dog or a cat. The results are shown in the contingency table. $$\begin{array}{lcc}\hline & \text { Dog } & \text { Cat } \\\\\hline \text { Female } & 16 & 41 \\ \text { Male } & 53 & 11 \\\\\hline\end{array}$$ Answer the following questions. Round to the nearest whole percentage as necessary. What percentage in the survey either are female or own a dog?

Consider the set of bivariate data shown in the table, and compute the indicated sums. $$\begin{array}{cccccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ y & 5 & 16.5 & 20 & 21.2 & 21.8 & 22.1 & 22.4 \\ \hline \end{array}$$ $$(\Sigma y)^{2}-\Sigma y^{2}$$

Consider the bivariate data shown in the table, and compute the indicated sums. $$\begin{array}{cccccccccc} \hline x & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ y & 10.2 & 9.3 & 8.8 & 7.1 & 6.6 & 5.3 & 4.2 & 3 & 2.9 \\ \hline \end{array}$$ $$\Sigma x \Sigma y$$
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