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Chapter 1: Problem 61

Use set-builder notation to write each set. The set of all even numbers between 9 and 99

Short Answer

Expert verified
\(\{ x \mid x \text{ is an even number and } 10 \leq x \leq 98 \}\)

Step by step solution

01

Identify the Range of Numbers

Determine the range of numbers given in the problem. Here, the range is all numbers between 9 and 99, not including 9 and 99.
02

Determine the Type of Numbers Needed

The exercise specifies even numbers within the range. Even numbers are those that are divisible by 2.
03

Express the Condition in Set-Builder Notation

In set-builder notation, describe the set including the condition that numbers are even and within the range. Use the expression: \(\{ x \mid x \text{ is an even number and } 10 \leq x \leq 98 \}\).

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Key Concepts

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

Even Numbers
Even numbers are integers that can be divided by 2 without leaving a remainder. In simpler terms, they are numbers like 2, 4, 6, and so on. Any number ending in 0, 2, 4, 6, or 8 is an even number.

To check if a number is even:
  • Divide the number by 2.
  • If the result is an integer (i.e., no remainder), it's even.
In our exercise, we needed all the even numbers between 9 and 99. Examples of these even numbers are 10, 12, 14, ..., and 98.
Span your even number list properly to avoid missing any valid even number in the range.
Range of Numbers
The range of numbers refers to a set that includes all numbers between two given endpoints. The endpoint numbers themselves are not always included. In our exercise, the given range is between 9 and 99.
Let's break it down:
The endpoints here are 9 and 99, but because the range described is 'between 9 and 99,' we exclude 9 and 99 themselves.
So, our range actually starts at 10 and ends at 98.
Always pay attention to whether the endpoints are included or excluded when dealing with ranges.
Set Theory
Set theory is a branch of mathematical logic that studies sets, which are collections of objects. Sets are usually represented by curly braces { }. Each object in a set is called an element or a member.
In set-builder notation, we describe the properties of the elements in the set.
For this exercise, we used set-builder notation to represent all the even numbers between 9 and 99. This can be written as:
\( \{ x \mid x \text{ is an even number and } 10 \leq x \leq 98\} \)
Here:
  • \( x \) represents an element in the set.
  • The pipe symbol (\( \mid \)) means 'such that.'
  • The rest of the expression describes the properties of \( x \): that it is an even number and it lies between 10 and 98 inclusive.
Understanding these notations is crucial for writing and interpreting mathematical sets correctly.

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Most popular questions from this chapter

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ \mathrm{N} \subseteq W $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ -(-3)^{2} $$

Classify each statement as either true or false. The following sets are used: \(\mathrm{N}=\) the set of natural numbers; \(W=\) the set of whole numbers; \(\mathbb{Z}=\) the set of integers; \(\mathbb{C}=\) the set of rational numbers; \(\mathrm{H}=\) the set of irrational numbers; \(\mathrm{R}=\) the set of real numbers. $$ \mathrm{W} \subseteq Z $$

Write an equivalent expression using a commutative law. Answers may vary. $$ 4 a+7 b $$

Jasmine is paid \(\$ 500\) per week plus \(10 \%\) of all sales she makes. a) Let \(x\) represent the amount of Jasmine's weekly sales, in dollars, and \(y\) her weekly paycheck. Write an equation expressing \(y\) in terms of \(x\) b) One week, Jasmine earned \(\$ 900 .\) What were her sales that week?
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